Structure and automorphisms of primitive coherent configurations
aa r X i v : . [ m a t h . C O ] A ug Structure and automorphisms of primitive coherentconfigurations ∗ Xiaorui Sun † Columbia University John Wilmes ‡ University of Chicago
Abstract
Coherent configurations (CCs) are highly regular colorings of the set ofordered pairs of a “vertex set”; each color represents a “constituent digraph.”CCs arise in the study of permutation groups, combinatorial structures suchas partially balanced designs, and the analysis of algorithms; their historygoes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituentdigraphs are connected.We address the problem of classifying PCCs with large automorphismgroups. This project was started in Babai’s 1981 paper in which he showedthat only the trivial PCC admits more than exp( e O ( n / )) automorphisms.(Here, n is the number of vertices and the e O hides polylogarithmic factors.)In the present paper we classify all PCCs with more than exp( e O ( n / )) automorphisms, making the first progress on Babai’s conjectured classifica-tion of all PCCs with more than exp( n ǫ ) automorphisms.A corollary to Babai’s 1981 result solved a then 100-year-old problem onprimitive but not doubly transitive permutation groups, giving an exp( e O ( n / )) bound on their order. In a similar vein, our result implies an exp( e O ( n / )) upper bound on the order of such groups, with known exceptions. This im-provement of Babai’s result was previously known only through the Clas-sification of Finite Simple Groups (Cameron, 1981), while our proof, likeBabai’s, is elementary and almost purely combinatorial.Our analysis relies on a new combinatorial structure theory we developfor PCCs. In particular, we demonstrate the presence of “asymptotically uni-form clique geometries” on PCCs in a certain range of the parameters. ∗ An extended abstract of this paper appeared in the Proceedings of the 47th ACM Symposiumon Theory of Computing (STOC’15) under the title
Faster canonical forms for primitive coherentconfigurations . † [email protected]. This work was partially supported by a grant from theSimons Foundation ( ‡ [email protected]. Research supported in part by NSF grant DGE-1144082. Introduction
Let V be a finite set; we call the elements of V “vertices.” A configuration ofrank r is a coloring c : V × V → { , . . . , r − } such that (i) c ( u, u ) = c ( v, w ) for any v = w , and (ii) for all i < r there is i ∗ < r such that c ( u, v ) = i iff c ( v, u ) = i ∗ . The configuration is coherent (CC) if (iii) for all i, j, k < r there isa structure constant p ijk such that if c ( u, v ) = i , there are exactly p ijk vertices w such that c ( u, w ) = j and c ( w, v ) = k . The diagonal colors c ( u, u ) are the vertexcolors , and the off-diagonal colors are the edge colors . A CC is homogeneous (HCC) if (iv) there is only one vertex color. We denote by R i the set of orderedpairs ( u, v ) of color c ( u, v ) = i . The directed graph X i = ( V, R i ) is the color- i constituent digraph . An HCC is primitive (PCC) if each constituent digraph isstrongly connected. An association scheme is an HCC for which i = i ∗ for allcolors i (so the constituent graphs X i are viewed as undirected).The term “coherent configuration” was coined by Donald Higman in 1969 [17],but the essential objects are older. In the case corresponding to a permutationgroup, CCs already effectively appeared in Schur’s 1933 paper [24]. This group-theoretic perspective on CCs was developed further by Wielandt [28].CCs appeared for the first time from a combinatorial perspective in a 1952paper by Bose and Shimamoto [10]. They, along with many of the subsequentauthors, consider the case of an association scheme, which is essential for under-standing partially balanced incomplete block designs, of interest to statisticians andto combinatorial design theorists. The generalization of an association scheme toan HCC was considered by Nair in 1964 [21]. The algebra associated with a CC,which already appeared in Schur’s paper, was rediscovered in 1959 in the contextof association schemes by Bose and Mesner [9].Weisfeiler and Leman [27] and Higman [17] independently defined CCs intheir full generality, including the associated algebra (called “cellular algebras” byWeisfeiler and Leman), in the late 1960s. For Higman, CCs were a generaliza-tion of permutation groups, whereas Weisfeiler and Leman were motivated by thealgorithmic Graph Isomorphism problem. In the intervening years, CCs, and as-sociation schemes in particular, have become basic objects of study in algebraiccombinatorics [8, 11, 7, 29]. CCs also continue to play a role in the study ofpermutation groups [16, 20]. Recent algorithmic applications of the CC conceptinclude the Graph Isomorphism problem [3] and the complexity of matrix multi-plication [15].PCCs are in a sense the “indivisible objects” among CCs and are therefore ofparticular interest.In this paper we classify the PCCs with the largest automorphism groups, upto the threshold stated in the following theorem. (See Defintion 1.3 and Theo-2em 1.4 for a more detailed statement, and see Section 1.6 for an explanation of theasymptotic notation used throughout, including O , e O , Θ , Ω , ∼ , and o .) Theorem 1.1. If X is a PCC not belonging to any of three exceptional families,then | Aut( X ) | ≤ exp( O ( n / log / n )) . Primitive permutation groups of large order were classified by Cameron [12].We refer to the orbital, or Schurian configurations of these groups as “Cameronschemes” (see Sections 1.1 and 1.2). For every ǫ > , for every n there is only abounded number of primitive groups of order greater than exp( n ǫ ) (the bound de-pends on ǫ but not on n ); we refer to this stratification as the “Cameron hierarchy.”Theorem 1.1 represents progress on the following conjectured classification ofPCCs with large automorphism groups. Conjecture 1.2 (Babai) . For every ε > , there is some N ε such that if X is a PCCon n ≥ N ε vertices and | Aut( X ) | ≥ exp( n ε ) , then X is a Cameron scheme. Inparticular, Aut( X ) is a known primitive group. This conjecture would be a far-reaching combinatorial generalization of Cameron’sclassification of large primitive permutation groups. In particular, while Cameron’sresult is only known through the Classification of Finite Simple Groups (CFSG),Conjecture 1.2 would imply (at least for orders greater than exp( n ǫ ) ) a CFSG-freeproof of Cameron’s result, giving a different kind of insight into the structure oflarge primitive permutation groups.Babai [1] established Conjecture 1.2 for all ε > / (the “first level of theCameron hierarchy”). As a corollary, he solved a then 100-year-old problem onprimitive but not doubly transitive permutation groups, giving an nearly tight, exp( e O ( n / )) bound on their order. The tight bound was subsequently found byCameron, using CFSG; our result implies a CFSG-free proof of the same tightbound. Moreover, our Theorem 1.1 confirms the conjecture to all ε > / , the firstimprovement since Babai’s paper. An elementary proof of an exp( e O ( n / )) upperbound on the order of primitive permutation groups, with known exceptions (the“second level of the Cameron hierarchy”) follows.For the proof of Theorem 1.1, we find new combinatorial structure in PCCs,including “clique geometries” in certain parameter ranges (Theorem 2.4). Anoverview of our structural results for PCCs is given in Section 2.Our motivation is thus twofold. First, we develop a structure theory for PCCs,the most general objects in a hierarchy of much-studied highly regular combinato-rial structures. Second, as a corollary to our main result, we obtain a CFSG-freeproof for the second level of the Cameron hierarchy of large primitive permutationgroups. 3dditional motivation for our work comes from the algorithmic Graph Isomor-phism problem. We explain this connection in Section 1.5. We now give a precise statement of our main combinatorial results.Given a graph X = ( V, E ) , we associate with X the configuration X ( X ) =( V ; ∆ , E, E ) where E denotes the set of edges of the complement of X . (We omit E if E = ∅ and omit E if E = ∅ .) So graphs can be viewed as configurations ofrank ≤ .Given an (undirected) graph H , the line-graph L ( H ) has as vertices the edgesof H , with two vertices adjacent in L ( H ) if the corresponding edges are incidentin H . The triangular graph T ( m ) is the line-graph of the complete graph K m (so n = (cid:0) m (cid:1) ). The lattice graph L ( m ) is the line-graph of the complete bipartitegraph K m,m (on equal parts) (so n = m ). The configurations X ( T ( m )) and X ( L ( m )) are coherent, and in fact primitive for m > . Definition 1.3.
A PCC is exceptional if it is of the form X ( X ) , where X is iso-morphic to the complete graph K n , the triangular graph T ( m ) , or the lattice graph L ( m ) , or the complement of such a graph.We note that the exceptional PCCs have n Ω( √ n ) automorphisms. Indeed, theexceptional PCCs are exactly the “orbital configurations” of large primitive permu-tation groups, as explained below.Our main result is that all the non-exceptional PCCs have far fewer automor-phisms. Theorem 1.4. If X is a non-exceptional PCC, then | Aut( X ) | ≤ exp( O ( n / log / n )) . We remark that the bound of Theorem 1.4 is tight, up to polylogarithmic factorsin the exponent. Indeed, the Johnson scheme J ( m, and the Hamming scheme H (3 , m ) both have exp(Θ( n / log n )) automorphisms. (The Johnson scheme J ( m, has vertices the -subsets of a domain of size m and c ( A, B ) = | A \ B | ,and the Hamming scheme H (3 , m ) has vertices the words of length from analphabet of size m with color c given by the Hamming distance.)The exceptional PCCs correspond naturally to the largest primitive permutationgroups. Given a permutation group G ≤ Sym( V ) , we define the orbital configura-tion X ( G ) on vertex set V with the R i given by the orbitals of G , i.e., the orbits ofthe induced action on V × V . CCs of this form were first considered by Schur [24],and are commonly called Schurian . Note that G ≤ Aut( X ( G )) .The Schurian CC X ( G ) is homogeneous if and only if G is transitive, andprimitive if and only if G is a primitive permutation group. If G is doubly transitive,4hen X ( G ) = X ( K n ) . We also have X ( S (2) m ) = X ( T ( m )) and X ( S m ≀ S ) = X ( L ( m )) . Following the completion of the Classification of Finite Simple Groups (CFSG),one of the tasks has been to obtain elementary proofs of results currently knownonly through CFSG. One such result is Cameron’s classification of all primitivepermutation groups of large order, obtained by combining CFSG with the O’Nan–Scott theorem [12]. Cameron’s threshold for the order is n O (log log n ) , but we stateMar´oti’s refinement of his classification of permutation groups of order greater than n c log n [18]. Theorem 1.5 (Cameron, Mar´oti) . If G is a primitive permutation group of degree n > , then one of the following holds:(a) there are positive integers d , k , and m such that ( A ( k ) m ) d ≤ G ≤ S ( k ) m ≀ S d ;(b) | G | ≤ n n . We call the primitive groups G of Theorem 1.5 (a) Cameron groups . Given aCameron group G with parameters d and k bounded, we obtain a PCC X ( G ) withexponentially large automorphism group H ≥ G , in particular, of order | H | ≥ exp( n / ( kd ) ) . We call the PCCs X ( G ) Cameron schemes when G is a Camerongroup.Hence, Conjecture 1.2 states that Cameron’s classification of primitive per-mutation groups transfers to the combinatorial setting of PCCs. Furthermore,the conjecture entails Cameron’s theorem, above the threshold | G | ≥ exp( n ε ) (see [5]). Hence, confirmation of Conjecture 1.2 would yield a CFSG-free proof ofCameron’s classification (above this threshold).It seems unlikely that combinatorial methods will match Cameron’s n O (log log n ) threshold for classification of primitive permutation groups. An n O (log n ) thresh-old (as in stated Theorem 1.5) via elementary techniques might be possible, sinceabove this threshold the socle of a primitive permutation group is a direct productof alternating groups, whereas below this threshold, simple groups of Lie type mayappear in the socle.However, until the present paper, the only CFSG-free classification of the largeprimitive permutation groups was given by Babai in a pair of papers in 1981 and1982 [1, 2]. Babai proved that | G | ≤ exp( O ( n / log n )) for primitive groups G other than A n and S n [1]. A corollary of our work gives the first CFSG-freeimprovement to Babai’s bound, by proving that | G | ≤ exp( O ( n / log / n )) for5rimitive permutation groups G , other than groups belonging to three exceptionalfamilies.In the following corollary to Theorem 1.1, S (2) m and A (2) m denote the actions of S m and A m , respectively, on the (cid:0) m (cid:1) pairs, and G ≀ H denotes the wreath productof the permutation groups G ≤ S n by H ≤ S m in the product action on a domainof size n m . Corollary 1.6.
Let Γ be a primitive permutation group of degree n . Then either | Γ | ≤ exp( O ( n / log / n )) , or Γ is one of the following groups:(a) S n or A n ;(b) S (2) m or A (2) m , where n = (cid:0) m (cid:1) ;(c) a subgroup of S m ≀ S containing ( A m ) , where n = m . The slightly stronger bound | Γ | ≤ exp( O ( n / log n )) follows from CFSG [12].By contrast, our proof is elementary.For given n = m , there are exactly three primitive groups in the third categoryof Corollary 1.6. We note that the groups of categories 1–3 of the corollary haveorder exp(Ω( n / log n )) .Corollary 1.6 follows from Theorem 1.4 by classifying the large primitivegroups G for which X ( G ) is an exceptional PCC, as in the following proposi-tion. Proposition 1.7.
There is a constant c such that the following holds. Let G ≤ S n be primitive, and suppose | G | ≥ n c log n .1. If X ( G ) = X ( K n ) , then G belongs to category (a) of Corollary 1.6.2. If X ( G ) = X ( T ( m )) , then G belongs to category (b) of Corollary 1.6.3. If X ( G ) = X ( L ( m )) , then G belongs to category (c) of Corollary 1.6. Proposition 1.7 as stated requires CFSG, but an elementary proof is availableunder the weaker bound of | G | ≥ exp( c log n ) using [23]. For the proof and amore general classification, we refer the reader to [5]. We now introduce the individualization/refinement heuristic. We shall use individ-ualization/refinement to find bases of automorphism groups of configurations.A base for a group G acting on a set V is a subset S ⊆ V such that thepointwise stablizer G ( S ) of S in G is trivial. If S is a base, then | G | ≤ | V | | S | .6et Iso( X , Y ) denote the set of isomorphisms from X to Y , and Aut( X ) =Iso( X , X ) . Individualization means the assignment of individual colors to some vertices;then the irregularity so created propagates via some canonical color refinementprocess. For a class C of configurations (not necessarily coherent), an assignment X X ′ is a color refinement if X , X ′ ∈ C have the same set of vertices andthe coloring of X ′ is a refinement of the coloring of X . Such an assignment is canonical if for all X , Y ∈ C , we have Iso( X , Y ) = Iso( X ′ , Y ′ ) . In particular, Aut( X ) = Aut( X ′ ) .Repeated application of the refinement process leads to the stable refinement after at most n − rounds.If after individualizing the elements of a set S ⊆ V , all vertices get differentcolors in the resulting stable refinement, we say that S completely splits X (withrespect to the given canonical refinement process). If S completely splits X , then S is a base for Aut( X ) . Hence, to prove Theorem 1.4, it suffices to show that some setof O ( n / log / n ) vertices completely splits X after canonical color refinement.For our purposes, the simple “naive vertex refinement” will suffice as ourcolor refinement procedure. Under naive vertex refinement , the edge-colors donot change, only the vertex-colors are refined. The refined color of vertex u of theconfiguration X encodes the following information: the current color of u and thenumber of vertices v of color i such that c ( u, v ) = j , for every pair ( i, j ) , where i is a vertex-color and j is an edge-color.We now state our main technical result, from which Theorem 1.4 immediatelyfollows. Theorem 1.8 (Main) . Let X be a non-exceptional PCC. Then there exists a set of O ( n / log / n ) vertices that completely splits X under naive refinement. This improves the main result of [1], which stated that if X is a PCC otherthan X ( K n ) , then there is a set of O ( n / log n ) vertices which completely splits X under naive refinement.Naive vertex refinement is the only color refinement used in the present paper.However, we remark that coherent configurations were first studied by Weisfeilerand Leman in the context of their stronger canonical color refinement [27, 26].Given a configuration X , the Weisfeiler–Leman (WL) canonical refinement pro-cess [27, 26] produces a CC X ′ on the same vertex set with Aut( X ) = Aut( X ′ ) ,by refining the coloring until it is coherent. More precisely, in every round of therefinement process, the color c ( u, v ) of the pair u, v ∈ V is replaced with a color c ′ ( u, v ) which encodes c ( u, v ) along with, for every pair j, k of original colors, thenumber of vertices w such that c ( u, w ) = i and c ( w, v ) = k . This refinement isiterated until the coloring stabilizes, i.e., the rank no longer increases in subsequent7ounds of refinement. The stable configurations under WL refinement are exactlythe coherent configurations. An undirected graph X = ( V, E ) is called strongly regular (SRG) with parameters ( n, k, λ, µ ) if X has n vertices, every vertex has degree k , each pair of adjacentvertices has λ common neighbors, and each pair of non-adjacent vertices has µ common neighbors.We note that a graph X is a SRG if and only if the configuration X ( X ) iscoherent. If a SRG X is nontrivial, i.e., it is connected and coconnected, then X ( X ) is a PCC.All of our exceptional PCCs are in fact SRGs. Our classification of PCCs, The-orem 1.4, was established in the special case of SRGs by Spielman in 1996 [25],on whose results we build. In fact, Chen, Sun, and Teng have now establisheda stronger bound for SRGs: a non-exceptional SRG has at most exp( e O ( n / )) automorphisms [14].The results of Spielman and Chen, Sun, and Teng both rely on Neumaier’sstructure theory [22] of SRGs to separate the exceptional SRGs with many au-tomorphisms from those to which I/R can be effectively applied. However, nogeneralization of Neumaier’s results to PCCs has been known. We provide a weakgeneralization, sufficient for our purposes, in Section 2. The “Graph Isomorphism (GI) problem” is the computational problem to decidewhether or not a pair of given graphs are isomorphic. This problem is of greatinterest to complexity theory since it is one of a very small number of naturalproblems in NP of intermediate complexity status (unlikely to be NP-complete butnot known to be solvable in polynomial time).In recent major development, Babai [3] announced a quasipolynomial-time( exp( O ((log n ) c )) ) algorithm.Babai’s algorithm reduces the problem to the isomorphism problem of PCC’sand then uses his (rather involved) “split-or-Johnson” procedure for further reduc-tion.Babai conjectures that a considerably simpler algorithm might succeed; unlessthe PCC is a Cameron scheme, individualization of a small number of vertices maycompletely split the vertex set. This is a more explicit version of Conjecture 1.2.Our result proves that this is indeed the case if “small number” means e O ( n / ) ,8mproving Babai’s e O ( n / ) . We hope that further refinement of our structure the-ory will yield further progress in this direction. To interpret asymptotic inequalities involving the parameters of a PCC, we thinkof the PCC as belonging to an infinite family in which the asymptotic inequalitieshold.For functions f, g : N → R > , we write f ( n ) = O ( g ( n )) if there is someconstant C such that f ( n ) ≤ Cg ( n ) , and we write f ( n ) = Ω( g ( n )) if g ( n ) = O ( f ( n )) . We write f ( n ) = Θ( g ( n )) if f ( n ) = O ( g ( n )) and f ( n ) = Ω( g ( n )) .We use the notation f ( n ) = e O ( g ( n )) when there is some constant c such that f ( n ) = O ( g ( n )(log n ) c ) . We write f ( n ) = o ( g ( n )) if for every ε > , there issome N ε such that for n ≥ N ε , we have f ( n ) < εg ( n ) . We write f ( n ) = ω ( g ( n )) if g ( n ) = o ( f ( n )) . We use the notation f ( n ) ∼ g ( n ) for asymptotic equality,i.e., lim n →∞ f ( n ) /g ( n ) = 1 . The asymptotic inequality f ( n ) . g ( n ) means g ( n ) ∼ max { f ( n ) , g ( n ) } . Acknowledgements
The authors are grateful to L´aszl´o Babai for sparking our interest in the problemaddressed in this paper, providing insight into primitive coherent configurationsand primitive groups, uncovering a faulty application of previous results in an earlyversion of the paper, and giving invaluable assistance in framing the results.
To prove Theorem 1.8, we need to develop a structure theory of PCCs. Theoverview in this section highlights the main components of that theory.Throughout the paper, X will denote a PCC of rank r on vertex set V withstructure constants p ijk for ≤ i, j, k ≤ r − . We assume throughout that r > ,since the case r = 2 is the trivial case of X ( K n ) , listed as one of our exceptionalPCCs. We also assume without loss of generality that color corresponds to thediagonal, i.e., R = { ( u, u ) : u ∈ V } .For any color i in a PCC, we write n i = n i ∗ = p ii ∗ = p i ∗ i , the out-degree ofeach vertex in X i .We say that color i is dominant if n i ≥ n/ . Colors i with n i < n/ are nondominant . We call a pair of distinct vertices dominant (nondominant) when its9olor is dominant (nondominant, resp.). We say color i is symmetric if i ∗ = i . Notethat when color i is dominant, it is symmetric, since n i ∗ = n i ≥ n/ .Our analysis will divide into two cases, depending on whether or not there is adominant color. In fact, many of the results of this section will assume that there isan overwhelmingly dominant color i satisfying n i ≥ n − O ( n / ) . The reduction tothis case is accomplished via Lemma 3.1 of the next section. The main structuralresult used in its proof is Lemma 2.1 below, which gives a lower bound on thegrowth of “spheres” in a PCC.For a color i and vertex u , we denote by X i ( u ) the set of vertices v such that c ( u, v ) = i . We denote by dist i ( u, v ) the directed distance from u to v in the color- i constituent digraph X i , and we write dist i ( j ) = dist i ( u, v ) for any vertices u, v with c ( u, v ) = j . (This latter quantity is well-defined by the coherence of X .) The δ -sphere X ( δ ) i ( u ) in X i centered at u is the set of vertices v with dist i ( u, v ) = δ . Lemma 2.1 (Growth of spheres) . Let X be a PCC, let i, j ≥ be nondiagonalcolors, let δ = dist i ( j ) , and u ∈ V . Then for any integer ≤ α ≤ δ − , we have | X ( α +1) i ( u ) || X ( δ − α ) i ( u ) | ≥ n i n j . We note that Lemma 2.1 is straightforward when X i is distance-regular. In-deed, a significant portion of the difficulty of the lemma was in finding the correctgeneralization. Overview of proof of Lemma 2.1.
The bipartite subgraphs of X i induced on pairsof the form ( X j ( u ) , X k ( u )) , where j, k are colors and u is a vertex, are biregularby the coherence of X i . We exploit this biregularity to count shortest paths in X i between a carefully chosen subset of X ( δ − α ) i ( u ) and X j ( u ) , for an arbitrary vertex u . The details of the proof are given in Section 4.In the rest of the paper, we assume without loss of generality that n =max i n i . We write ρ = P i ≥ n i = n − n − . For the rest of the section,color will in fact be dominant. In fact, every theorem in the rest of this sectionwill state the assumption that ρ = o ( n / ) . Lemma 2.2 below demonstrates someof the power of this supposition. Lemma 2.2.
Let ε > and let X be a PCC with ρ < (1 − ε ) n / ) . Then, for n sufficiently large, dist i (1) = 2 for every nondominant color i . Consequently, n i ≥ √ n − for i = 0 . verview of proof of Lemma 2.2. We will prove that if dist i (1) ≥ for some color i , then ρ & n / . Without loss of generality, we assume n ∼ n , since otherwisewe are already done.Fix an arbitrary vertex u and consider the bipartite graph B between X ( δ − i ( u ) and X ( u ) , with an edge from x ∈ X ( δ − i ( u ) to y ∈ X ( u ) when c ( x, y ) = i . Bythe coherence of X , the bipartite graph is regular on X ( u ) ; call its degree γ . Anobstacle to our analysis is that the graph need not be biregular. Nevertheless, weestimate the maximum degree β of a vertex in X ( δ − i ( u ) in B . We first note that n γ ≤ βρ .Let w be a vertex satisfying c ( u, w ) = i . We pass to the subgraph B ′ inducedon ( X ( δ − i ( w ) , X ( δ − i ( w )) , and observe that the degree of vertices in X ( δ − i ( u ) ∩ X ( δ − i ( w ) is preserved, while the degree of vertices in X ( u ) ∩ X ( δ − i ( w ) doesnot increase. Let v be a vertex of degree β in B ′ , and let j = c ( w, v ) . We finallyconsider the bipartite graph B ′′ on ( X j ( w ) , X w ) , where X w is the set of vertices x ∈ X ( δ − i ( w ) with at most γ in-neighbors in X i lying in the set X j ( w ) . Inparticular, X ( u ) ∩ X ( δ − i ( w ) ⊆ X w . This graph B ′′ is now regular (of degree ≥ β ) on X j ( w ) . Since X w ⊆ X ( δ − i ( w ) , we have | X w | ≤ ρ , which eventuallygives the bound β ≤ γρ /n . Combining this with our earlier estimate βρ ≥ n γ proves the lemma.The details of the proof are given in Section 6. Notation.
Let G ( X ) be the graph on V formed by the nondominant pairs. So G ( X ) is regular of valency ρ , and every pair of distinct nonadjacent vertices in G ( X ) has exactly µ common neighbors, where µ = P i,j ≥ p ij . The graph G ( X ) is not generally SR, since pairs of adjacent vertices in G ( X ) of different colors in X will in general have different numbers of common neighbors. However, intuitionfrom SRGs will prove valuable in understanding G ( X ) .We write N ( u ) for the set of neighbors of u in the graph G ( X ) . For i nondom-inant, we define λ i = | X i ( u ) ∩ N ( v ) | , where c ( u, v ) = i . So, the parameters λ i are loosely analogous to the parameter λ of a SRG.A clique C in an undirected graph G is a set of pairwise adjacent vertices; its order | C | is the number of vertices in the set. Definition 2.3. A clique geometry on a graph G is a collection G of maximalcliques such that every pair of adjacent vertices in G belongs to a unique cliquein G . A clique geometry of a PCC X is a clique geometry on G ( X ) . The cliquegeometry G is asymptotically uniform (for an infinite family of PCCs) if for every C ∈ G , u ∈ C , and nondominant color i , we have either | C ∩ X i ( u ) | ∼ λ i or | C ∩ X i ( u ) | = 0 (as n → ∞ ). 11e have the following sufficient condition for the existence of clique geome-tries in PCCs. Theorem 2.4.
Let X be a PCC satisfying ρ = o ( n / ) , and fix a constant ε > . If λ i ≥ εn / for every nondominant color i , then for n sufficiently large, there is aclique geometry G on X . Moreover, G is asymptotically uniform. Theorem 2.4 provides a powerful dichotomy for PCCs: either there is an upperbound on some parameter λ i , or there is a clique geometry. Adapting a philosophyexpressed in [4], we note that bounds on λ i are useful because they limit the corre-lation between the i -neighborhoods of two random vertices. Similar bounds on theparameter λ of a SRG were used in [4].On the other hand, Theorem 2.4 guarantees that if all parameters λ i are suffi-ciently large, the PCC has an asymptotically uniform clique geometry. This is ourweak analogue of Neumaier’s geometric structure. Clique geometries offer theirown dichotomy. Geometries with at most two cliques at a vertex can classified;this includes the exceptional PCCs (Theorem 2.5 below). A far more rigid struc-ture emerges when there are at least three cliques at every vertex. In this case,we exploit the ubiquitous 3-claws (induced K , subgraphs) in G ( X ) in order toconstruct a set which completely splits X (Lemma 3.4 (b)). Overview of proof of Theorem 2.4.
The existence of a weaker clique structure fol-lows from a result of Metsch [19]. (See Lemma 7.1 below and the comments inthe paragraph preceding it.) Specifically, under the hypotheses of Theorem 2.4, forevery nondominant color i and vertex u , there is a partition of X i ( u ) into cliquesof order ∼ λ i in G ( X ) . We call such a collection of cliques a local clique partition (referring to the color- i neighborhood of any fixed vertex).The challenge is to piece together these local clique partitions into a cliquegeometry. An obstacle is that Metsch’s cliques are cliques of G ( X ) , not X i ; thatis, the edges of the cliques partitioning X i ( u ) have nondominant colors but not ingeneral color i . In particular, for two vertices u, v ∈ V with c ( u, v ) = i , the cliquecontaining v in the partition of X i ( u ) may not correspond to any of the cliques inthe partition of X i ( v ) .We first generalize these local structures. An I -local clique partition is a parti-tion of S i ∈ I X i ( u ) into cliques of order ∼ P i ∈ I λ i . We study the maximal sets I for which such I -local clique partitions exist, and eventually prove that these max-imal sets I partition the set of nondominant colors, and the corresponding cliquesare maximal in G ( X ) .Finally, we prove a symmetry condition: given a nondominant pair of vertices u, v ∈ V , the maximal local clique at u containing v is equal to the maximal12ocal clique at v containing u . This symmetry ensures the cliques form a cliquegeometry, and this clique geometry is asymptotically uniform by construction.The details of the proof are given in Section 7.The case that X has a clique geometry with some vertex belonging to at mosttwo cliques includes the exceptional CCs corresponding to T ( m ) and L ( m ) . Wegive the following classification. Theorem 2.5.
Let X be a PCC such that ρ = o ( n / ) . Suppose that X has anasymptotically uniform clique geometry G and a vertex u ∈ V belonging to atmost two cliques of G . Then for n sufficiently large, one of the following is true:(a) X has rank three and is isomorphic to X ( T ( m )) or X ( L ( m )) ;(b) X has rank four, X has a non-symmetric non-dominant color i , and G ( X ) isisomorphic to T ( m ) for m = n i + 2 .Overview of proof of Theorem 2.5. We first use the coherence of X to show thatevery vertex u ∈ V belongs to exactly two cliques of G , and these cliques haveorder ∼ ρ/ . By counting vertex-clique incidences, we then obtain the estimate ρ . √ n . On the other hand, by Lemma 2.2, every nondominant color i satisfies n i & √ n . Hence, there are at most nondominant colors.Since every vertex belongs to exactly two cliques, the graph G ( X ) is the line-graph of a graph. If there is only one nondominant color, then G ( X ) is stronglyregular, and therefore, for n sufficiently large, G ( X ) is isomorphic to T ( m ) or L ( m ) . On the other hand, if there are two nondominant colors, by counting pathsof length we show that G ( X ) must again be isomorphic to T ( m ) . By studyingthe edge-colors at the intersection of the cliques containing two distinct verticesand exploiting the coherence of X , we finally eliminate the case that the two non-dominant colors are symmetric.The details of the proof are given in Section 8. We now give a high-level overview of how we apply our structure theory of PCCsto prove Theorem 1.8.Most of the results highlighted in Section 2 assumed that ρ = o ( n / ) . Hence,the first step is to reduce to this case, which we accomplish via the followinglemma. Lemma 3.1.
Let X be a PCC. If ρ ≥ n / (log n ) − / , then there is a set of size O ( n / (log n ) / ) which completely splits X .
13e remark that in the case that the rank r of X is bounded, our Lemma 3.1 fol-lows from a theorem of Babai [1, Theorem 2.4]. Following Babai [1], we analyzethe distinguishing number. Definition 3.2.
Let u, v ∈ V . We say w ∈ V distinguishes u and v if c ( w, u ) = c ( w, v ) . We write D ( u, v ) for the set of vertices w distinguishing u and v , and D ( i ) = | D ( u, v ) | where c ( u, v ) = i . We call D ( i ) the distinguishing number of i .Hence, D ( i ) = P j = k p ijk ∗ . If w ∈ D ( u, v ) , then after individualizing w andrefining, u and v get different colors.Babai observed that in order to completely split a PCC X , it suffices to indi-vidualize some set of O ( n log n/D min ) vertices, where D min = min i =0 { D ( i ) } [1,Lemma 5.4]. Thus, to prove Lemma 3.1, we show that if ρ ≥ n / (log n ) − / thenfor every color i = 0 , we have D ( i ) = Ω( n / (log n ) − / ) .The following bound on the number of large colors in a PCC becomes powerfulwhen D ( i ) is small. Lemma 3.3.
Let X be a PCC. For any nondiagonal color i , the number of colors j such that n j > n i / is at most O ((log n + n/ρ ) D ( i ) /n i ) .Overview of proof of Lemma 3.3. Let I α be the set of colors i such that D ( i ) ≤ α ,and J β the set of colors j such that n j ≤ β . For a set I of colors, let n I = P i ∈ I n i be the total degree of the colors in I .First, we prove that ⌊ α/ (3 D ( i )) ⌋ n i ≤ n I α , a lower bound on the total degreeof colors with distinguishing number ≤ α . Next, we prove a lemma that allows usto transfer estimates for the total degree of colors with small distinguishing numberinto estimates for the total degree of colors with low degree. Specifically, we provethat n J β ≤ α , where β = n I α / . Together, these two results allow us to transferestimates on total degree between the sets I α and J β , as α and β increase.The details of the proof are given in Section 5. Overview of proof of Lemma 3.1.
Fix a color i ≥ . We wish to give a lowerbound on D ( i ) . Babai observed that for any color j ≥ , we have D ( i ) ≥ D ( j ) / dist i ( j ) [1, Proposition 6.4 and Theorem 6.1]). Hence, we wish to givean upper bound on dist i ( j ) for some color j with D ( j ) large.We analyze two cases: n / (log n ) − / ≤ ρ < n/ and ρ ≥ n/ .In the former case, when n / (log n ) − / ≤ ρ < n/ , we first observe that D (1) = Ω( ρ ) . Hence, the problem is reduced to bounding the quantity dist i (1) forevery color i . Our bound in Lemma 2.1 on the size of spheres suffices for this tasksince n is large.In the case that ρ ≥ n/ , Babai observed that the color j maximizing D ( j ) sat-isfies D ( j ) = Ω( n ) . We partition the colors of X according to their distinguishing14umber, by first partitioning the positive integers less than D ( j ) into cells of length D ( i ) . (Specifically, we partition the colors of X so that the α -th cell contains thecolors k satisfying D ( i ) α ≤ D ( k ) < D ( i )( α + 1) , and there are O ( D ( j ) /D ( i )) cells.) Each cell of this partition P is nonempty. In fact, we show that the sum ofthe degrees of the colors in each cell is at least n i .On the other hand, Lemma 3.3 says that there are few colors k satisfying n k >n i / , and we show that the total degree of the colors k with n k ≤ n i / is alsosmall. Since each cell of the partition has degrees summing to at least n i , thesetogether give an upper bound on the number of cells, and hence a lower bound on D ( i ) .The details of the proof are given in Section 5.We have now reduced to the case that ρ = o ( n / ) . Our analysis of this case isinspired by Spielman’s analysis of SRGs [25]. Lemma 3.4.
There exists a constant ε > such that the following holds. Let X be a PCC with ρ = o ( n / ) . If X satisfies either of the following conditions, thenthere is a set of O ( n / (log n ) / ) vertices which completely splits X .(a) There is a nondominant color i such that λ i < εn / .(b) For every nondominant color i , we have λ i ≥ εn / . Furthermore, X has anasymptotically uniform clique geometry C such that every vertex belongs toat least three cliques of C .Overview of proof of Lemma 3.4. We will show that if we individualize a randomset of O ( n / (log n ) / ) vertices, then with positive probability, every pair of dis-tinct vertices gets different colors in the stable refinement.Let u, v ∈ C , and fix two colors i and j . Generalizing a pattern studied bySpielman, we say a triple ( w, x, y ) is good for u and v if c ( u, x ) = c ( u, y ) = c ( x, y ) = 1 , c ( u, w ) = i , and c ( w, x ) = c ( w, y ) = j , but there exists no vertex z such that c ( v, z ) = i and c ( z, x ) = c ( z, y ) = j . (See Figure 3). To ensure that u and v get different colors in the stable refinement, it suffices to individualize twovertices x, y ∈ V for which there exists a vertex w such that ( w, x, y ) is good for u and v . We show that if there are many good triples for u and v , then individualizinga random set of O ( n / (log n ) / ) vertices is overwhelming likely to result in theindividualization of such a pair x, y ∈ V .Condition (a) of the lemma is analogous to the asymptotic consequences ofNeumaier’s claw bound used by Spielman [25] (cf. [6, Section 2.2]), except that thebound on λ i does not imply a similar bound on λ i ∗ . We show that a relatively weakbound on λ i ∗ already suffices for Spielman’s argument to essentially go through.15owever, if even this weaker assumption fails, then we turn to our local cliquestructure for the analysis (as described in the overview of Theorem 2.4).When condition (b) holds, we cannot argue along Spielman’s lines, and insteadanalyze the structural properties of our clique geometries to estimate the numberof good triples.The details of the proof are given in Section 9.By Theorem 2.4, either the hypotheses of of Lemma 3.4 are satisfied, or X has an asymptotically uniform clique geometry C , and some vertex belongs to atmost two cliques of C . Theorem 2.5 gives a characterization PCCs X with thelatter property: X is one of the exceptional PCCs, or X has rank four with a non-symmetric non-dominant color i and G ( X ) is isomorphic to T ( m ) for m = n i + 2 .We handle this final case via the following lemma, proved in Section 8. Lemma 3.5.
Let X be a PCC satisfying Theorem 2.5 (b). Then some set of size O (log n ) completely splits X . We conclude this overview by observing that Theorem 1.8 follows from theabove results.
Proof of Theorem 1.8.
Let X be a PCC. Suppose first that ρ ≥ n / (log n ) − / .Then by Lemma 3.1, there is a set of size O ( n / (log n ) / ) which completelysplits X .Otherwise, ρ < n / (log n ) − / = o ( n / ) . By Theorem 2.4, either the hy-potheses of Lemma 3.4 are satisfied, or the hypotheses of Theorem 2.5 are satisfied.In the former case, some set of O ( n / (log n ) / ) vertices completely splits X . Inthe latter case, either X is exceptional, or, by Lemma 3.5, some set of O (log n ) vertices completely splits X . In this section, we will prove Lemma 2.1, our estimate of the size of spheres inconstituent digraphs.We start from a few basic observations.
Proposition 4.1.
Let G = ( A, B, E ) be a bipartite graph, and let A ∪ · · · ∪ A m bea partition of A such that the subgraph induced on ( A i , B ) is biregular of positivevalency for each ≤ i ≤ m . Then for any A ′ ⊆ A , we have | N ( A ′ ) | / | A ′ | ≥ | B | / | A | where N ( A ′ ) is the set of neighbors of vertices in A ′ , i.e., N ( A ′ ) = { y ∈ B : ∃ x ∈ A ′ , { x, y } ∈ E } . roof. Let A ′ ⊆ A . By the pigeonhole principle, there is some i such that | A ′ ∩ A i | / | A i | ≥ | A ′ | / | A | . Let α be the degree of a vertex in A i and let β be the numberof neighbors in A i of a vertex in B . We have α | A i | = β | B | , and β | N ( A ′ ∩ A i ) | ≥ α | A ′ ∩ A i | . Hence, | N ( A ′ ) | ≥ | N ( A ′ ∩ A i ) | ≥ | A ′ ∩ A i | αβ = | A ′ ∩ A i || B || A i | ≥ | B || A ′ || A | . Suppose
A, B ⊆ V are disjoint set of vertices. We denote by ( A, B, i ) thebipartite graph between A and B such that there is an edge from x ∈ A to y ∈ B if c ( x, y ) = i . For I ⊆ [ r − a set of nondiagonal colors, we denote by ( A, B, I ) thebipartite graph between A and B such that there is an edge from x ∈ A to y ∈ B if c ( x, y ) ∈ I . Fact 4.2.
For any vertex u , colors ≤ j, k ≤ r − with j = k , and set I ⊆ [ r − of nondiagonal colors, the bipartite graph ( X j ( u ) , X k ( u ) , I ) is biregular.Proof. The degree of every vertex in X j ( u ) is P i ∈ I p jik ∗ . And the degree of everyvertex in X k ( u ) is P i ∈ I p kji .Recall our notation X ( δ ) i ( u ) for the δ -sphere centered at u in the color- i con-stituent digraph, i.e., the set of vertices v such that dist i ( u, v ) = δ .For the remainder of Section 4, we fix a PCC X , a color ≤ i ≤ r − , and avertex u . For a color ≤ j ≤ r − and an integer ≤ α ≤ dist i ( j ) , we denoteby S ( j ) α the set of vertices v ∈ X ( α ) i ( u ) such that there is a vertex w ∈ X j ( u ) and ashortest path in X i from u to w passing through v , i.e., S ( j ) α = { v ∈ X ( α ) i ( u ) : ∃ w ∈ X j ( u ) s.t. dist i ( u, v ) + dist i ( v, w ) = dist i ( u, w ) } . Note that these sets S ( j ) α are nonempty by the primitivity of X , and in particular, if α = dist i ( j ) , then S ( j ) α = X j ( u ) . For v ∈ V and an integer dist i ( u, v ) < α ≤ dist i ( j ) , we denote by S ( j ) α ( v ) ⊆ S ( j ) α the set of vertices x ∈ S ( j ) α such that thereis a shortest path in X i from u to x passing through v , i.e. S ( j ) α ( v ) = S ( j ) α ∩ X ( α − dist i ( u,v )) i ( v )= { x ∈ S ( j ) α : dist i ( u, v ) + dist i ( v, x ) = dist i ( u, x ) } . See Figure 1 for a graphical explanation of the notation.
Corollary 4.3.
Let ≤ j ≤ r − be a color such that δ = dist i ( j ) ≥ . Let ≤ α ≤ δ − be an integer, and let v ∈ S ( j ) α . Then | S ( j ) δ ( v ) || S ( j ) α +1 ( v ) | ≥ n j | S ( j ) α +1 | . w v. . . . . . X (1) i ( u ) . . . . . . X ( α ) i ( u ) S ( j ) α X ( α +1) i ( u ) S ( j ) α +1 ( v ) S ( j ) α +1 X j ( u ) Figure 1: S ( j ) α and S ( j ) α +1 ( v ) . Proof.
Consider the bipartite graph ( S ( j ) α +1 , X j ( u ) , I ) with I = { k : 1 ≤ k ≤ r − and dist i ( k ) = dist i ( j ) − α − } . There is an edge from x ∈ S ( j ) α +1 to y ∈ X j ( u ) if there is a shortest path from u to y passing through x .By the coherence of X , if X ℓ ( u ) ∩ S ( j ) α +1 is nonempty for some color ℓ , then X ℓ ( u ) ⊆ S ( j ) α +1 . Hence, S ( j ) α +1 is partitioned into sets of the form X ℓ ( u ) with dist i ( ℓ ) = α + 1 . For such colors ℓ , by Fact 4.2, ( X ℓ ( u ) , X j ( u ) , I ) is biregular,and by the definition of S ( j ) α +1 , then ( X ℓ ( u ) , X j ( u ) , I ) is not an empty graph.Therefore, the result follows by applying Proposition 4.1 with A = S ( j ) α +1 , B = X j ( u ) , A ′ = S ( j ) α +1 ( v ) ⊆ S ( j ) α +1 , and (hence) N ( A ′ ) = S ( j ) δ ( v ) . Fact 4.4.
Let ≤ j ≤ r − be a color such that δ = dist i ( j ) ≥ , and w be a vertex in X j ( u ) . Let ≤ α ≤ δ − , and let v be a vertex in S ( j ) α . If dist i ( v, w ) = δ − α , then { x : x ∈ X i ( v ) and dist i ( x, w ) = δ − α − } ⊆ S ( j ) α +1 ( v ) . Proof.
For any x ∈ X i ( v ) , we have dist i ( u, x ) ≤ α +1 . If dist i ( x, w ) = δ − α − ,then x ∈ X ( α +1) i ( u ) , because otherwise dist( u, w ) < δ . Then x is in S ( j ) α +1 ( v ) ,since there is a shortest from u to w passing through x .18 roposition 4.5. Let ≤ j ≤ r − be a color such that δ = dist i ( j ) ≥ . Let ≤ α ≤ δ − , and let v ∈ S ( j ) α . Then | X δ − αi ( u ) | ≥ n i | S ( j ) δ ( v ) || S ( j ) α +1 ( v ) | . Proof.
Let k be a color satisfying dist i ( k ) = δ − α and X k ( v ) ∩ S ( j ) δ ( v ) = ∅ . Let w be a vertex in X k ( v ) ∩ S ( j ) δ ( v ) . Consider the bipartite graph B = ( X i ( v ) , X k ( v ) , I ) ,where I = { ℓ : dist i ( ℓ ) = δ − α − } .By Fact 4.2, B is biregular, and by Fact 4.4 the degree of w in B is at most | S ( j ) α +1 ( v ) | . Denote by d k the degree of a vertex x ∈ X i ( v ) in B , so n k | S ( j ) α +1 ( v ) | ≥ n i d k . Hence, summing over all colors k such that X k ( v ) ∩ S ( j ) δ ( v ) = ∅ , we have | X ( δ − α ) i ( v ) | ≥ X k n k ≥ X k n i d k | S ( j ) α +1 ( v ) | ≥ n i | S ( j ) δ ( v ) || S ( j ) α +1 ( v ) | . Finally, by the coherence of X , we have | X ( δ − α ) i ( u ) | = | X ( δ − α ) i ( v ) | .We now complete the proof of Lemma 2.1. Proof of Lemma 2.1.
Combining Corollary 4.3 and Proposition 4.5, for any ≤ α ≤ δ − we have | X ( δ − α ) i ( u ) | ≥ n i n k | S ( k ) α +1 | and so since S ( k ) α +1 ⊆ X ( α +1) i ( u ) by definition, we have the desired inequality. In this section, we will prove Lemma 3.1, which will allow us to assume that ourPCCs X satisfy ρ = o ( n / ) .We recall that the distinguishing number D ( i ) of a color i is the number ofvertices w such that c ( w, u ) = c ( w, v ) , where u and v are any fixed pair of ver-tices such that c ( u, v ) = i . Hence, D ( i ) = P k = j p ijk ∗ . If D ( i ) is large for everycolor i > , then for every pair of distinct vertices u, v ∈ V , a random individual-ized vertex w gives different colors to u and v in the stable refinement with goodprobability. This idea is formalized in the following lemma due to Babai [1]. Lemma 5.1 (Babai [1, Lemma 5.4]) . Let X be a PCC and let ζ = min { D ( i ) : 1 ≤ i ≤ r − } . Then there is a set of size O ( n log n/ζ ) whichcompletely splits X .
19e give the following lower bound on ζ when ρ is sufficiently large. Lemma 5.2.
Let X be a PCC and suppose that ρ ≥ n / (log n ) − / .Then D ( i ) = Ω( n / (log n ) − / ) for all ≤ i ≤ r − . Lemma 3.1 follows immediately from Lemmas 5.1 and 5.2.We will prove Lemma 5.2 by separately addressing the cases ρ ≥ n/ and ρ < n/ . The case ρ < n/ will rely on our estimate for the size of spheres in con-stituent digraphs, Lemma 2.1. For the case ρ ≥ n/ , we will rely on Lemma 3.3,which bounds the number of large colors when D ( i ) is small for some color i ≥ .We prove Lemma 3.3 in the following subsection.We first recall the following observation of Babai [1, Proposition 6.3]. Proposition 5.3 (Babai) . Let X be a PCC. Then n − r − X j =1 D ( j ) n j ≥ ρ + 2 . The following corollary is then immediate.
Corollary 5.4.
Let X be a PCC. There exists a nondiagonal color i with D ( i ) > ρ . The following facts about the parameters of a coherent configuration are stan-dard.
Proposition 5.5 ([29, Lemma 1.1.1, 1.1.2, 1.1.3]) . Let X be a CC. Then for allcolors i, j, k , the following relations hold:(i) n i = n i ∗ (ii) p ijk = p i ∗ k ∗ j ∗ (iii) n i p ijk = n j p jik ∗ (iv) P r − j =0 p ijk = P r − j =0 p ikj = n k We now prove Lemma 3.3, using the following preliminary results.
Lemma 5.6.
Let X be a PCC, let I be a nonempty set of nondiagonal colors, let n I = P i ∈ I n i , and let J be the set of colors j such that n j ≤ n I / . Then X j ∈ J n j ≤ { D ( i ) : i ∈ I } . roof. For any color i , by Proposition 5.5, we have D ( i ) = r − X j =0 X k = j p ijk ∗ = r − X j =0 X k = j n j p jik n i = 1 n i r − X j =0 n j X k = j p jik = 1 n i r − X j =0 n j ( n i − p jij ) . Therefore, n I max { D ( i ) : i ∈ I } ≥ X i ∈ I n i D ( i ) ≥ X i ∈ I X j ∈ J n j ( n i − p jij ) ≥ X j ∈ J n j X i ∈ I ( n i − p jij ) ≥ X j ∈ J n j ( n I − n j ) ≥ n I X j ∈ J n j . Lemma 5.7.
Let X be a PCC, and suppose p ijk > for some i, j, k . Then D ( j ) − D ( k ) ≤ D ( i ) ≤ D ( j ) + D ( k ) . Proof.
Fix vertices u, v, w ∈ V with c ( u, w ) = i , c ( u, v ) = j , and c ( v, w ) = k .(These vertices exist since p ijk > .) For any vertex x such that c ( x, u ) = c ( x, w ) ,we have c ( x, u ) = c ( x, v ) or c ( x, v ) = c ( x, w ) . Therefore, D ( j ) + D ( k ) ≥ D ( i ) .For the other inequality, if p ijk > then p jik ∗ > by Proposition 5.5, and D ( k ∗ ) = D ( k ) by the definition of distinguishing number. So we have D ( i ) + D ( k ) = D ( i ) + D ( k ∗ ) ≥ D ( j ) , using the previous paragraph for the latter in-equality. Lemma 5.8.
Let X be a PCC. Then for any nondiagonal color i and number ≤ η ≤ ρ − D ( i ) , there is a color j such that η < D ( j ) ≤ η + D ( i ) .Proof. By Corollary 5.4, there is a color k with D ( k ) > ρ . Now consider a shortestpath u , . . . , u ℓ in X i with c ( u , u ℓ ) = k . (By the primitivity of X , the digraph X i is21trongly connected, and such a path exists.) Let δ j = D ( c ( u , u j )) for ≤ j ≤ k .By Lemma 5.7, we have | δ j − δ j +1 | ≤ D ( i ) . Hence, one of the numbers δ j falls inthe interval ( η, η + D ( i )] for any ≤ η ≤ ρ − D ( i ) .We denote by I α the set of colors i with D ( i ) ≤ α . Lemma 5.9.
Let X be a PCC with ρ > . Let i be a nondiagonal color and let ≤ η ≤ ρ − D ( i ) . Then n i ≤ X j ∈ I η +3 D ( i ) \ I η n j . Proof.
By Lemma 5.8, the set I η +2 D ( i ) \ I η + D ( i ) is nonempty. Let k ∈ I η +2 D ( i ) \ I η + D ( i ) . We have P r − j =0 p kij = n i by Proposition 5.5. On the other hand, if p kij > for some j , then D ( j ) − D ( i ) ≤ D ( k ) ≤ D ( j ) + D ( i ) by Lemma 5.7, and so j ∈ I η +3 D ( i ) \ I η . Hence, n i = r − X j =0 p kij = X j ∈ I η +3 D ( i ) \ I η p kij ≤ X j ∈ I η +3 D ( i ) \ I η n j . Lemma 5.10.
Let X be a PCC with ρ > , let i be a nondiagonal color, and let ≤ η ≤ ρ . Then (cid:22) η D ( i ) (cid:23) n i ≤ X j ∈ I η n j . Proof. If η < D ( i ) , the left-hand side is , so assume η ≥ D ( i ) . For any integer ≤ α ≤ ⌊ η/ (3 D ( i )) ⌋ , let S α = I D ( i ) α \ I D ( i )( α − . Then ⌊ η/ (3 D ( i )) ⌋ [ α =1 S α ⊆ I η By the disjointness of the sets S α and Lemma 5.9, we have X j ∈ I η n j ≥ ⌊ η/ (3 D ( i )) ⌋ X α =1 X j ∈ S α n j ≥ (cid:22) η D ( i ) (cid:23) n i . Finally, we are able to prove Lemma 3.3.22 roof of Lemma 3.3.
Fix an integer ≤ α ≤ ⌊ log ( ρ/ (3 D ( i ))) ⌋ . For any number β , let J β denote the set of colors j such that n j ≤ β . We start by estimating | J α n i \ J α − n i | , i.e., the number of colors j with α − n i < n j ≤ α n i . ByLemma 5.10, we have X j ∈ I α (3 D ( i )) n j ≥ α n i . Therefore, applying Lemma 5.6 with I = I α (3 D ( i )) and J = J α n i , we have X j ∈ J αni n j ≤ { D ( i ) : i ∈ I α · D ( i ) } ≤ α +1 (3 D ( i )) , with the second inequality coming from the definition of I α (3 D ( i )) .It follows that the number of colors j such that j ∈ J α n i \ J α − n i is at most α +1 (3 D ( i )) / (2 α − n i ) = 12 D ( i ) /n i . Overall, the number of colors j satisfying (1 / n i < n j ≤ ⌊ log ( ρ/ D ( i )) ⌋ n i is at most n + 1) D ( i ) /n i .Furthermore, the number of colors j satisfying n j > ⌊ log ( ρ/ D ( i )) ⌋ n i ≥ ρn i D ( i ) is at most (6 D ( i ) / ( ρn i )) n , since P r − j =0 n j = n . Hence, the number of colors j such that n j > n i / is at most O ((log n + n/ρ ) D ( i ) /n i ) . We now prove Lemma 5.2, our lower bound for D ( i ) .First, we recall the following two observations made by Babai [1, Proposition6.4 and Theorem 6.11]. Proposition 5.11 (Babai) . Let X be a PCC. For colors ≤ i, j ≤ r − , we have D ( j ) ≤ dist i ( j ) D ( i ) . Proposition 5.12 (Babai) . Let X be a PCC. For any color ≤ i ≤ r − , we have n i D ( i ) ≥ n − . We prove the following two estimates of the distinguish number23 emma 5.13.
Let X be a PCC. Fix nondiagonal colors i, j ≥ and a vertex u ∈ V . Let δ = dist i ( j ) , and γ = P δ − α =2 | X ( α ) i ( u ) | . If δ ≥ , then D ( i ) = Ω (cid:18) D ( j ) √ nn j γ (cid:19) / ! . Proof.
By Lemma 2.1, for any ≤ α ≤ δ − we have | X ( α +1) i ( u ) || X ( δ − α ) i ( u ) | ≥ n i n j and in particular, max {| X ( α +1) i ( u ) | , | X ( δ − α ) i ( u ) |} ≥ √ n i n j . Hence, γ = δ − X α =2 | X ( α ) i ( u ) | = Ω( δ √ n i n j ) = Ω δ √ nn j p D ( i ) ! , (1)where the last inequality comes from Proposition 5.12. Now by Proposition 5.11and Eq. (1), we have D ( i ) ≥ D ( j ) δ = Ω D ( j ) √ nn j γ p D ( i ) ! , from which the desired inequality immediately follows. Lemma 5.14.
Let X be a PCC with ρ = Ω( n ) . Then every nondiagonal color i with n i ≤ ρ satisfies D ( i ) = Ω (cid:18)r ρn i log n (cid:19) . Proof.
Fix a nondiagonal color i with n i ≤ ρ , and suppose D ( i ) < ρ/ (otherwisethe lemma holds trivially). Let J β denote the set of colors j such that n j ≤ β .Applying Lemma 5.6 with the set I = { i } , we have X j ∈ J ni/ n j ≤ D ( i ) . (2)On the other hand, by Lemma 5.9, for every integer η with ≤ η ≤ ρ/ − D ( i ) , n i ≤ X j ∈ I η +3 D ( i ) \ I η n j . Thus, for every such η , at least one of following two conditions hold:24a) there exists a color j ∈ I η +3 D ( i ) \ I η satisfying n j > n i / ;(b) X j ∈ I η +3 D ( i ) \ I η : n j ≤ n i / n j ≥ n i .There are at least ⌊ ρ/ (6 D ( i )) ⌋ disjoint sets of the form I η +3 D ( i ) \ I η with ≤ η ≤ ρ/ − D ( i ) . By Lemma 3.3, at most O ((log n + n/ρ ) D ( i ) /n i ) = O ((log n ) D ( i ) /n i ) of these satisfy (a). By Eq. (2), at most D ( i ) /n i satisfy (b).Hence, ⌊ ρ/ (6 D ( i )) ⌋ = O ((log n ) D ( i ) /n i ) , giving the desired inequality.We recall that when color is dominant, it is symmetric. In this case, werecall our notation µ = | N ( x ) ∩ N ( y ) | , where x, y ∈ V are any pair of verticeswith c ( x, y ) = 1 and N ( x ) is the nondominant neighborhood of x . Hence, µ = P i,j> p ij . Lemma 5.15.
Let X be a PCC with n ≥ n/ . Then µ ≤ ρ /n .Proof. Fix a vertex u . There are at most ρ paths of length two from u along edgesof nondominant color, and exactly n vertices v such that c ( u, v ) = 1 . For anysuch vertex y , there are exactly µ paths of length two from u to v along edges ofnondominant color. Hence, µ ≤ ρ /n . Proof of Lemma 5.2.
First, suppose n / (log n ) − / ≤ ρ < n/ . We have n = n − ρ − > n/ − . Consider two vertices u, v ∈ V with c ( u, v ) = 1 . Note thatfor any vertex w ∈ N ( v ) \ N ( u ) , we have c ( w, u ) = 1 and c ( w, v ) > . Hence,by Lemma 5.15 and the definition of D (1) , D (1) ≥ ρ − µ ≥ ρ − ρ n ≥ (cid:18) − o (1) (cid:19) ρ = Ω( n / (log n ) − / ) . Fix a color i = 1 . If dist i (1) = 2 , then by Proposition 5.11, D ( i ) ≥ D (1)2 ≥ Ω( n / (log n ) / ) . Otherwise, if dist i (1) ≥ , by applying Lemma 5.13 with j = 1 , we have D ( i ) = Ω (cid:18) D (1) √ nn n − n (cid:19) / ! = Ω (cid:18) ρnρ − (cid:19) / ! = Ω( n / ) . Now suppose ρ ≥ n/ . By Lemma 5.14 and Proposition 5.12, for every color i with n i ≤ ρ , we have ( D ( i )) / = Ω s ρn i D ( i )log n ! = Ω (cid:18)r ρn log n (cid:19) , D ( i ) = Ω( n / (log n ) − / ) . If n ≤ ρ , then n i ≤ ρ for all i ,and we are done. Otherwise, if n > ρ , we have only to verify that D (1) =Ω( n / (log n ) − / ) . Consider two vertices u, w with dist ( u, w ) = 2 . (Since weassume the rank is at least , we can always find such u, w by the primitivity of X .)Let i = c ( u, w ) . Then i > and so n i ≤ ρ . Since D ( i ) = Ω( n / (log n ) − / ) forevery color < i ≤ r − , and dist ( i ) = 2 , we have D (1) = Ω( n / (log n ) − / ) by Proposition 5.11. We now prove Lemma 2.2, which states that dist i (1) = 2 for any nondominantcolor i , assuming that inequality ρ = o ( n / ) .We start from a few basic observations. Observation 6.1.
Let X be a PCC. For any nondominant color i , we have n i ≥ n /ρ .Proof. Fix a vertex u ∈ V . Since X i is connected ( X is primitive), for any v ∈ X ( x ) , there is a shortest path in X i from u to v , hence there exists a vertex w ∈ N ( u ) such that c ( w, v ) = i , so | N ( x ) | n i ≥ | X ( x ) | . Lemma 6.2.
Let X be a PCC with a nondominant color i , let δ = dist ( i ) , andsuppose δ ≥ . Any vertices u, w with c ( u, w ) = i satisfy the following twoproperties:(i) If v ∈ X ( δ − i ( u ) ∩ X ( δ − i ( w ) , then X i ( v ) ∩ X ( u ) ⊆ X i ( v ) ∩ X ( δ − i ( w ); (ii) If z ∈ X ( u ) ∩ X ( δ − i ( w ) , then X i ∗ ( z ) ∩ X ( δ − i ( w ) ⊆ X i ∗ ( z ) ∩ X ( δ − i ( u ) . Proof. If v ∈ X ( δ − i ( u ) ∩ X ( δ − i ( w ) then dist i ( u, v ) = δ − and dist i ( w, v ) = δ − . So for any vertex x ∈ X i ( v ) , we have dist i ( w, x ) ≤ δ − . If x ∈ X ( u ) ,then dist i ( w, x ) = δ − , since otherwise dist i ( u, x ) < δ .Similarly, z ∈ X ( u ) ∩ X ( δ − i ( w ) means dist i ( u, z ) = δ and dist i ( w, z ) = δ − . So for any y satisfying dist i ( w, y ) = δ − , we have dist i ( u, y ) ≤ δ − . If z ∈ X i ( y ) , then dist i ( u, y ) = δ − , since otherwise dist i ( u, z ) < δ .26 emma 6.3. Let X be a PCC with a nondominant color i , let δ = dist ( i ) ,and suppose δ ≥ . Fix a vertex u ∈ V and let B be the bipartite graph ( X ( δ − i ( u ) , X ( u ) , i ) . Let γ denote the minimum degree in B of a vertex in X ( u ) ,and let β denote the maximum degree in B of a vertex in X ( δ − i ( u ) . Then β ≤ γρ /n .Proof. In fact, B is regular on X ( u ) by Fact 4.2, so every vertex in X ( u ) hasdegree γ in B .Let v ∈ X ( δ − i ( u ) achieve degree β in B , and let w ∈ X i ( u ) be such that dist i ( w, v ) = δ − . Let B ′ be the subgraph of B given by ( S , S , i ) , where S = X ( δ − i ( u ) ∩ X ( δ − i ( w ) and S = X ( u ) ∩ X ( δ − i ( w ) . Note that v ∈ S .By Lemma 6.2 (i), every neighbor of v in B is also a neighbor of v in B ′ , and inparticular, the degree of v in B ′ is again β .Let j = c ( w, v ) , and let H = ( X j ( w ) , S , i ) , where S = n z ∈ X ( δ − i ( w ) : | X i ∗ ( z ) ∩ X j ( w ) | ≤ γ o . Recall that v ∈ X j ( w ) , so v is also a vertex of H . We claim that every neighbor of v in B ′ is also a neighbor of v in H , so the degree of v in B ′ is again ≥ β . Indeed,let z ∈ X i ( v ) ∩ S . Then z ∈ X ( δ − i ( w ) , and furthermore | X i ∗ ( z ) ∩ X j ( w ) | ≤ | X i ∗ ( z ) ∩ X ( δ − i ( w ) | ≤ | X i ∗ ( z ) ∩ X ( δ − i ( u ) | = γ. So, z ∈ S , and every neighbor of v in B ′ is also a vertex of H as claimed.Now by Fact 4.2, H is regular on X j ( w ) with degree ≥ β . Hence, βn j ≤ | E ( H ) | ≤ γ | S | ≤ γ | X ( δ − i ( w ) | ≤ γρ. The lemma follows by Observation 6.1.
Proof of Lemma 2.2.
We prove that if dist i (1) ≥ for some color i , then ρ & n / . Without loss of generality, we assume n ∼ n , since otherwise we arealready done.Let i be such that dist i (1) ≥ , and write δ = dist i (1) . Fix a vertex u and let B , γ , and β be as in Lemma 6.3, so β ≤ γρ /n . By Lemma 6.3, B is regular on X ( u ) . Let γ, β be defined as Lemma 6.3. By Lemma 6.3, we have β ≤ γρ /n .Therefore, by counting the number of edges in Bρ γn ≥ β | X ( δ − i ( x ) | ≥ | E ( B ) | = n γ. The lemma is then immediate since n ∼ n .27 Clique geometries
In this section, we prove Theorem 2.4, giving sufficient conditions for the existenceof an asymptotically uniform clique geometry in a PCC.We use the word “geometry” in Definition 2.3 because the cliques resemblelines in a geometry: two distinct cliques intersect in at most one vertex. Indeed, aregular graph G has a clique geometry G with cliques of uniform order only if it isthe point-graph of a geometric 1-design with lines corresponding to cliques of G .Theorem 2.4 builds on earlier work of Metsch [19] on the existence of similarclique structures in “sub-amply regular graphs” (cf. [6]) via the following lemma.The lemma can be derived from [19, Theorem 1.2], but see [6, Lemma 4] for aself-contained proof. Lemma 7.1.
Let H be a graph on k vertices which is regular of degree λ and suchthat any pair of nonadjacent vertices have at most µ common neighbors. Supposethat kµ = o ( λ ) . Then there is a partition of V ( H ) into maximal cliques of order ∼ λ , and all other maximal cliques of H have order o ( λ ) . Metsch’s result, applied to the graphs induced by G ( X ) on sets of the form X i ( u ) , gives collections of cliques which locally resemble asymptotically uniformclique geometries. These collections satisfy the following definition for a set I = { i } containing a single color. Definition 7.2.
Let I be a set of nondominant colors. An I -local clique partition ata vertex u is a collection P of subsets of X I ( u ) satisfying the following properties:(i) P is a partition of X I ( u ) into maximal cliques in the subgraph of G ( X ) induced on X I ( u ) ;(ii) for every C ∈ P u and i ∈ I , we have | C ∩ X i ( u ) | ∼ λ i .We say X has I -local clique partitions if there is an I -local clique partition at everyvertex u ∈ V .To prove Theorem 2.4, we will stitch local clique partitions together into geo-metric clique structures.Note that from the definition, if P is an I -local clique partition (at some vertex)and i ∈ I , then |P| ∼ n i /λ i . Corollary 7.3.
Let X be a PCC and let i be a nondominant color such that n i µ = o ( λ i ) . Then X has { i } -local clique partitions.Proof. Fix a vertex u , and apply Lemma 7.1 to the graph H induced by G ( X ) on X i ( u ) . The Lemma gives a collection of cliques satisfying Definition 7.2.28he following simple observation is essential for the proofs of this section. Observation 7.4.
Let X be a PCC, let C be a clique in G ( X ) , and suppose u ∈ V \ C is such that | N ( u ) ∩ C | > µ . Then C ⊆ N ( u ) .Proof. Suppose there exists a vertex v ∈ C \ N ( u ) , so c ( u, v ) = 1 . Then | N ( u ) ∩ N ( v ) | = µ by the definition of µ in a PCC. But | N ( u ) ∩ N ( v ) | ≥ | N ( u ) ∩ C ∩ N ( v ) | = | N ( u ) ∩ ( C \{ v } ) | = | N ( u ) ∩ C | > µ, a contradiction.Under modest assumptions, if local clique partitions exist, they are unique. Lemma 7.5.
Let X be a PCC, let i be a nondominant color such that n i µ = o ( λ i ) ,and let I be a set of nondominant colors such that i ∈ I . Suppose X has I -localclique partitions. Then for every vertex u ∈ V , there is a unique I -local cliquepartition P at u .Proof. Let u ∈ V and let P be an I -local clique partition at u . Let C and C ′ be two distinct maximal cliques in the subgraph of G ( X ) induced on X I ( u ) . Weshow that | C ∩ C ′ | < µ . Suppose for the contradiction that | C ∩ C ′ | ≥ µ . Fora vertex v ∈ C \ C ′ , we have | N ( v ) ∩ ( C ′ ∪ { u } ) | > µ , and so C ′ ⊆ N ( v ) byObservation 7.4. But since v / ∈ C ′ , this contradicts the maximality of C ′ . So infact | C ∩ C ′ | < µ .Now let C / ∈ P be a maximal clique in the subgraph of G ( X ) induced on X I ( u ) . Since P is an I -local clique partition, it follows that | C | = X C ′ ∈P | C ′ ∩ C | < µ |P| ∼ n i µ/λ i = o ( λ i ) . Then C does not belong to an I -local clique partition, since it fails to satisfy Defi-nition 7.2 (ii). Suppose X has I -local clique partitions, and c ( u, v ) ∈ I for some u, v ∈ V . Weremark that in general, the clique containing v in the I -local clique partition at u will not be in any way related to any clique in the I -local clique partition at v .In particular, we need not have c ( v, u ) ∈ I . However, even when c ( v, u ) ∈ I aswell, there is no guarantee that the clique at u containing v will have any particular29elation to the clique at v containing u . This lack of symmetry is a fundamentalobstacle that we must overcome to prove Theorem 2.4.Lemma 7.7 below is the main result of this subsection. It gives sufficient con-ditions on the parameters of a PCC for finding the desired symmetry in local cliquepartitions satisfying the following additional condition. Definition 7.6.
Let I be a set of nondominant colors, let u ∈ V , and let P be an I -local clique partition at u . We say P is strong if for every C ∈ P , the clique C ∪ { u } is maximal in G ( X ) . We say X has strong I -local clique partitions if thereis a strong I -local clique partition at every vertex u ∈ V .We introduce additional notation. Suppose I is a set of nondominant colors,and i ∈ I satisfies n i µ = o ( λ i ) . If X has I -local clique partitions, then for every u, v ∈ V with c ( u, v ) ∈ I , we denote by K I ( u, v ) the set C ∪ { u } , where C is theclique in the partition of X I ( u ) containing v (noting that by Lemma 7.5, this cliqueis uniquely determined). Lemma 7.7.
Let X be a PCC with ρ = o ( n / ) , let i be a nondominant color,and let I and J be sets of nondominant colors such that i ∈ I , i ∗ ∈ J , and X has strong I -local and J -local clique partitions. Suppose λ i λ i ∗ = Ω( n ) . Then forevery u, v ∈ V with c ( u, v ) = i , we have K I ( u, v ) = K J ( v, u ) . We first prove two easy preliminary statements.
Proposition 7.8.
Suppose ρ = o ( n / ) . Then µ = o ( n / ) and µρ = o ( n ) .Furthermore, for every nondominant color i , we have µ = o ( n i ) .Proof. By Lemma 5.15, µ ≤ ρ /n = o ( n / ) , and then µρ = o ( n ) . The lastinequality follows by Lemma 2.2. Lemma 7.9.
Let X be a PCC and let I and J be sets of nondominant colors suchthat X has strong I -local and J -local clique partitions. Suppose that for somevertices u, v, x, y ∈ V we have | K I ( u, v ) ∩ K J ( x, y ) | > µ . Then K I ( u, v ) = K J ( x, y ) .Proof. Suppose there exists a vertex z ∈ K J ( x, y ) \ K I ( u, v ) . We have | N ( z ) ∩ K I ( u, v ) | ≥ | K J ( x, y ) ∩ K I ( u, v ) | > µ . Then K I ( u, v ) ⊆ N ( z ) by Observa-tion 7.4, contradicting the maximality of K I ( u, v ) . Thus, K J ( x, y ) ⊆ K I ( u, v ) .Similarly, K I ( u, v ) ⊆ K J ( x, y ) . Proof of Lemma 7.7.
Without loss of generality, assume λ i ≤ λ i ∗ .Suppose for contradiction that there exists a vertex u ∈ V such that for every v ∈ X i ( u ) , we have K I ( u, v ) = K J ( v, u ) . Then | K I ( u, v ) ∩ K J ( v, u ) | ≤ µ
30y Lemma 7.9. Fix v ∈ X i ( u ) , so for every w ∈ K I ( u, v ) ∩ X i ( u ) , we have | K J ( w, u ) ∩ K I ( u, v ) | ≤ µ . Hence, there exists some sequence w , . . . , w ℓ of ℓ = ⌈ λ i / (2 µ ) ⌉ vertices w α ∈ K I ( u, v ) ∩ X i ( u ) such that K J ( w α , u ) = K J ( w β , u ) for α = β . But by Lemma 7.9, for α = β we have | K J ( w α , u ) ∩ K J ( w β , u ) | ≤ µ .Hence, for any ≤ α ≤ ℓ we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K J ( w α , u ) \ [ β = α K J ( w β , u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & λ i ∗ − µλ i / (2 µ ) ≥ λ i ∗ / . But K J ( w α , u ) ⊆ N ( u ) , so | N ( u ) | ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ [ α =1 K J ( w α , u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & λ i λ i ∗ µ = ω ( ρ ) by Proposition 7.8. This contradicts the definition of ρ .Hence, for any vertex u , there is some v ∈ X i ( u ) such that K I ( u, v ) = K J ( v, u ) . Then, in particular, | X i ∗ ( v ) ∩ X I ( u ) | & λ i ∗ by the definition of a J -local clique partition. By the coherence of X , for every v ∈ X i ( u ) , we have | X i ∗ ( v ) ∩ X I ( u ) | & λ i ∗ . Recall that X I ( u ) is partitioned into ∼ n i /λ i maximalcliques, and for each of these cliques C other than K I ( u, v ) , we have | N ( v ) ∩ C | ≤ µ . Hence, | X i ∗ ( v ) ∩ K I ( u, v ) | & λ i ∗ − O (cid:18) µn i λ i (cid:19) = λ i ∗ − o (cid:18) nλ i (cid:19) ∼ λ i ∗ by Proposition 7.8. Since the J -local clique partition at v partitions X i ∗ ( v ) into ∼ n i /λ i ∗ cliques, at least one of these intersects K I ( u, v ) in at least ∼ λ i ∗ /n i = ω ( µ ) vertices. In other words, there is some x ∈ X i ∗ ( v ) such that | K J ( v, x ) ∩ K I ( u, v ) | = ω ( µ ) . But then K J ( v, x ) = K I ( u, v ) by Lemma 7.9. In particu-lar, u ∈ K J ( v, x ) , so K J ( v, x ) = K J ( v, u ) . Hence, K J ( v, u ) = K J ( v, x ) = K I ( u, v ) , as desired. Our next step in proving Theorem 2.4 is showing the existence of strong localclique partitions. We accomplish this via the following lemma.
Lemma 7.10.
Let X be a PCC such that ρ = o ( n / ) , and let i be a nondominantcolor such that n i µ = o ( λ i ) . Suppose that for every color j with n j < n i , we have λ j = Ω( √ n ) . Then for n sufficiently large, there is a set I of nondominant colorswith i ∈ I such that X has strong I -local clique partitions.
31e will prove Lemma 7.10 via a sequence of lemmas which gradually improveour guarantees about the number of edges between cliques of the I -local cliquepartition at a vertex u and the various neighborhoods X j ( u ) for j / ∈ I . Lemma 7.11.
Let X be a PCC, and let i and j be nondominant colors. Then forany < ε < and any u, v ∈ V with c ( u, v ) = j , we have | X i ( u ) ∩ N ( v ) | ≤ max (cid:26) λ i + 11 − ε , n i r µεn j (cid:27) Proof.
Fix u, v ∈ V with c ( u, v ) = j and let α = | X i ( u ) ∩ N ( v ) | . We countthe number of triples ( x, y, z ) of vertices such that x, y ∈ X i ( u ) ∩ N ( z ) , with c ( u, z ) = j and c ( x, y ) = 1 . There are at most n i pairs x, y ∈ X i ( u ) , and if c ( x, y ) = 1 then there are at most µ vertices z such that x, y ∈ N ( z ) . Hence, thenumber of such triples is at most n i µ . On the other hand, by the coherence of X , forevery z with c ( u, z ) = j , we have at least α ( α − λ i − pairs x, y ∈ X i ( u ) ∩ N ( z ) with c ( x, y ) = 1 . Hence, there are at least n j α ( α − λ i − total such triples. Thus, n j α ( α − λ i − ≤ n i µ. Hence, if α ≤ ( λ i +1) / (1 − ε ) , then we are done. Otherwise, α > ( λ i +1) / (1 − ε ) ,and then λ i + 1 < (1 − ε ) α . So, we have n i µ ≥ n j α ( α − λ i − > εn j α , and then α < n i p µ/ ( εn j ) . Lemma 7.12.
Let X be a PCC, and let i be a nondominant color such that n i µ = o ( λ i ) . Let I be a set of nondominant colors with i ∈ I such that X has I -localclique partitions. Let j be a nondominant color such that n i p µ/n j < ( √ / λ i .Let u ∈ V , let P u be the I -local clique partition at u , and let v ∈ X j ( u ) . Supposesome clique C ∈ P u is such that c ( u, v ) = j and | N ( v ) ∩ C | ≥ µ . Then for everyvertex x, y ∈ V with c ( x, y ) = j , letting P x be the I -local clique partition at x ,the following statements hold:(i) there is a unique clique C ∈ P x such that C ⊆ N ( y ) ;(ii) | N ( y ) ∩ X i ( x ) | ∼ λ i .Proof. Letting b C = C ∪ { u } , we have | b C ∩ N ( v ) | ≥ µ + 1 > µ . Therefore, byObservation 7.4, we have C ⊆ N ( v ) . In particular, | N ( v ) ∩ X i ( u ) | & λ i , and so bythe coherence of X , | N ( y ) ∩ X i ( x ) | & λ i for every pair x, y ∈ V with c ( x, y ) = j .32ow fix x ∈ V , and let P x be the I -local clique partition at x . By the definitionof an I -local clique partition, we have |P x | ∼ n i /λ i . For every y ∈ X j ( x ) , byassumption we have | N ( y ) ∩ X i ( x ) | & λ i = ω ( µn i /λ i ) . (3)Then it follows from the pigeonhole principle that for n sufficiently large, thereis some clique C ∈ P x such that | N ( y ) ∩ C | > µ , and then C ⊆ N ( y ) byObservation 7.4.Now suppose for contradiction that there is some clique C ′ ∈ P x with C ′ = C ,such that C ′ ⊆ N ( y ) . | N ( y ) ∩ X i ( x ) | ≥ | C ∪ C ′ | & λ i ∼ λ i + 1) (4)(with the last relation holding since λ i = ω ( √ n i µ ) = ω (1) .) However, byLemma 7.11 with ε = 1 / , we have | X i ( x ) ∩ N ( y ) | ≤ max (
32 ( λ i + 1) , n i s µn j ) = 32 ( λ i + 1) , with the last equality holding by assumption. This contradicts Eq. (4), so we con-clude that C is the unique clique in P x satisfying C ⊆ N ( y ) . In particular, byObservation 7.4, we have | N ( y ) ∩ C ′ | ≤ µ for every C ′ ∈ P x with C ′ = C .Finally, we estimate | N ( y ) ∩ X i ( x ) | by | N ( y ) ∩ X i ( x ) ∩ C | + X C ′ = C | N ( y ) ∩ X i ( x ) ∩ C ′ | . λ i + µn i /λ i ∼ λ i , which, combined with Eq. (3), gives | N ( y ) ∩ X i ( x ) | ∼ λ i . Lemma 7.13.
Let X be a PCC, and let i be a nondominant color such that n i µ = o ( λ i ) . There exists a set I of nondominant colors with i ∈ I such that X has I -localclique partitions and the following statement holds. Suppose j is a nondominantcolor such that n i p µ/n j = o ( λ i ) , let u ∈ V , and let P be the I -local cliquepartition at u . Then for any C ∈ P and any vertex v ∈ X j ( u ) \ C , we have | N ( v ) ∩ C | < µ .Proof. By Corollary 7.3, X has { i } -local clique partitions. Let I be a maximalsubset of of the nondominant colors such that i ∈ I and X has I -local cliquepartitions. We claim that I has the desired property.33ndeed, suppose there exists some color j / ∈ I satisfying n i p µ/n j = o ( λ i ) ,some vertices u, v with c ( u, v ) = j , and some C ∈ P with | N ( v ) ∩ C | ≥ µ , where P is the I -local clique partition at u . By Lemma 7.12, for n sufficiently large, forevery vertex u, v ∈ V with c ( u, v ) = j , and I -local clique partition P at u , thereis a unique clique C ∈ P such that C ⊆ N ( v ) , and furthermore | N ( v ) ∩ X i ( u ) | ∼ λ i . (5)Now fix u ∈ V and let P be the I -local clique partition at u . Let P ′ be thecollection of sets C ′ of the form C ′ = C ∪ { v ∈ X j ( u ) : C ⊆ N ( v ) } for every C ∈ P . Let J = I ∪ { j } . We claim that P ′ satisfies Definition 7.2, so X has local clique partitions on J . This contradicts the maximality of I , and thelemma then follows.First we verify Definition 7.2 (i). By the second paragraph of this proof, P ′ partitions X J ( u ) . Furthermore, the sets C ∈ P ′ are cliques in G ( X ) , since for any C ∈ P ′ and any distinct v, w ∈ C ∩ X j ( u ) , we have | N ( v ) ∩ N ( w ) | ≥ | C ∩ X I ( u ) | & λ i = ω ( µ ) and so c ( v, w ) is nondominant by the definition of µ . Furthermore, the cliques C ∈ P ′ are maximal in the subgraph of G( X ) induced on X J ( u ) , since they aremaximal when restricted to X I ( u ) and each vertex v ∈ X j ( u ) extends a uniqueclique in the restriction of P ′ to X I ( u ) .We now verify Definition 7.2 (ii). By the pigeonhole principle, there is some C ∈ P ′ with | C ∩ X j ( u ) | & n j |P ′ | = n j |P| ∼ λ i n j n i . But since C is a clique in G ( X ) , we have | C ∩ X j ( u ) | ≤ λ j + 1 . So, from thedefining property of j , λ j + 1 & λ i n j n i = ω ( √ µn j ) Since n j and µ are positive integers, we have in particular λ j = ω (1) , and thus λ j & λ i n j n i = ω ( √ µn j ) . (6)Hence, n j µ = o ( λ j ) , and so by Corollary 7.3, X has { j } -local clique partitions.34et C ′ ⊆ X j ( u ) be a maximal clique in G ( X ) of order ∼ λ j . By Eq. (5) thereare ∼ λ j λ i nondominant edges between C ′ and X i ( u ) , so some x ∈ X i ( u ) satisfies | N ( x ) ∩ C ′ | & λ j λ i /n i = ω ( λ j q µ/n j ) = ω ( µ ) . (The last equality uses Eq. (6).) Furthermore, by Eq. (6), we have n j . ( n i /λ i ) λ j = o ( p n i /µλ j ) , where the last inequality comes from the assumption that √ n i µ = o ( λ i ) . So byapplying Lemma 7.12 with { j } in place of I , it follows that for every x ∈ X i ( u ) ,we have | N ( x ) ∩ X j ( u ) | ∼ λ j .We count the nondominant edges between X i ( u ) and X j ( u ) in two ways: thereare ∼ λ j such edges at each of the n i vertices in X i ( u ) , and (by Eq. (5)) there are ∼ λ i such edges at each of the n j vertices in X j ( u ) . Hence, n i λ j ∼ n j λ i .Now, using Eq. (6), µ |P ′ | ∼ µn i /λ i ∼ µn j /λ j = o ( λ j ) . By the maximality ofthe cliques C ∈ P ′ in the subgraph of G ( X ) induced on X J ( u ) , for every distinct C, C ′ ∈ P ′ and v ∈ C , we have | N ( v ) ∩ C ′ | ≤ µ . Therefore, for v ∈ C ∩ X j ( u ) ,we have λ j − | X j ( u ) ∩ C | = | X j ( u ) ∩ N ( v ) | − | X j ( u ) ∩ C |≤ | ( N ( v ) ∩ X j ( u )) \ C |≤ µ |P ′ | = o ( λ j ) , so that | X j ( u ) ∩ C | ∼ λ j , as desired.Now P ′ satisfies Definition 7.2, giving the desired contradiction. Proof of Lemma 7.10.
Suppose for contradiction that no set I of nondominant col-ors with i ∈ I is such that X has strong I -local clique partitions. Without lossof generality, we may assume that n i is minimal for this property, i.e., for everynondominant color j with n j < n i , there is a set J of nondominant colors with j ∈ J such that X has strong I -local clique partitions.Let I be the set of nondominant colors containing i guaranteed by Lemma 7.13.Let u ∈ V be such that some clique C in the I -local clique partition at u isnot maximal in G ( X ) . In particular, let v ∈ V \ C be such that C ⊆ N ( v ) ,and let j = c ( u, v ) . Then j is a nondominant color, and j / ∈ I . Furthermore,by the defining property of I (the guarantee of Lemma 7.13), it is not the casethat n i p µ/n j = o ( λ i ) . In particular we may take n j < n i , since otherwise, if n j ≥ n i , then n i p µ/n j ≤ √ n i µ = o ( λ i ) by assumption. Now since n j < n i ,also λ j = Ω( √ n ) by assumption. Furthermore, by the minimality of n i , there is35 set J of nondominant colors with j ∈ J such that X has strong J -local cliquepartitions on J . In particular, i / ∈ J .By the definition of I -local clique partitions, | N ( v ) ∩ X i ( u ) | ≥ | N ( v ) ∩ X i ( u ) ∩ C | & λ i . Now let D be the clique containing v in the J -local clique partition at u . By thecoherence of X , for every x ∈ X j ( u ) ∩ D , we have | N ( x ) ∩ X i ( u ) | & λ i . Hence,there are & λ j λ i nondominant edges between X j ( u ) ∩ D and X i ( u ) . So, by thepigeonhole principle, some vertex y ∈ X i ( u ) satisfies | N ( y ) ∩ D ∩ X j ( u ) | & λ i λ j n i = ω (cid:18)r µn i λ j (cid:19) = ω (cid:18)r µnn i (cid:19) = ω ( µ ) . (The second inequality uses the assumption that √ n i µ = o ( λ i ) . The last inequalityuses Proposition 7.8.) But then D \ { y } ⊆ N ( y ) by Observation 7.4. Then y ∈ D by the definition of a strong local clique partition, and so i ∈ J , a contradiction.We conclude that in fact X has strong I -local clique partitions.We finally complete the proof of Theorem 2.4. Proof of Theorem 2.4.
By Lemma 7.10, for every nondominant color i there is aset I such that X has strong local clique partitions on I . We claim that these sets I partition the collection of nondominant colors. Indeed, suppose that there are twosets I and J of nondominant colors such that i ∈ I ∩ J and X has strong I -local and J -local clique partitions. Let u, v ∈ V be such that c ( u, v ) = i . By the uniquenessof the induced { i } -local clique partition at u (Lemma 7.5), we have | K I ( u, v ) ∩ K J ( u, v ) | & λ i = ω ( µ ) , so K I ( u, v ) = K J ( u, v ) , and I = J . In particular, for every nondominant color i , there exists a unique set I of nondominant colors such that X has strong I -localclique partitions.We simplify our notation and write K ( u, v ) = K I ( u, v ) whenever c ( u, v ) ∈ I and X has strong I -local clique partitions. By Lemma 7.7, we have K ( u, v ) = K ( v, u ) for all u, v ∈ V with c ( u, v ) nondominant. Let G be the collection ofcliques of the form K ( u, v ) for c ( u, v ) nondominant. Then G is an asymptoticallyuniform clique geometry. 36 .3 Consequences of local clique partitions for the parameters λ i We conclude this section by analyzing some consequences for the parameters λ i ofour results on strong local clique partitions. Lemma 7.14.
Let X be a PCC with ρ = o ( n / ) . For every nondominant color i ,we have λ i < n i − .Proof. Suppose for contradiction that λ i = n i − for some nondominant color i .For every nondominant color j , by Proposition 7.8, we have n i p µ/n j = o ( n i ) = o ( λ i ) . Furthermore, n i µ = o ( λ i ) . Let I be the set of nondominant colors with i ∈ I guaranteed by Lemma 7.13. In particular, X has I -local clique partitions.In fact, since λ i = n i − , for every vertex u and every clique C in the I -localclique partition at u , we have C ∩ X i ( u ) ∼ n i . Hence, there is only one cliquein the I -local clique partition at u , and so X I ( u ) is a clique in G ( X ) . For everyvertex u , let K I ( u ) = X I ( u ) ∪ { u } . Then for every vertex v / ∈ X I ( u ) , we have | N ( v ) ∩ K I ( u ) | ≤ µ . In particular, K I ( u ) is a maximal clique in G ( X ) , and X hasstrong I -local clique partitions.Let U, v ∈ V with v ∈ X I ( u ) , let j = c ( u, v ) ∈ I , and suppose | K I ( v ) ∩ K I ( u ) | > µ . Then K I ( v ) = K I ( u ) by Lemma 7.9. Hence, by the coherence of X , for any w, x ∈ V with x ∈ X j ( w ) , K I ( w ) = K I ( x ) . By applying this factiteratively, we find that for any two vertices y, z ∈ V such that there exists a pathfrom y to z in X j , we have z ∈ K I ( y ) , contradicting the primitivity of X . Weconclude that | K I ( v ) ∩ K I ( u ) | ≤ µ if c ( u, v ) ∈ I . Hence, if we fix a vertex u andcount pairs of vertices ( v, w ) ∈ X i ( u ) × X I ( u ) with c ( w, v ) = i , we have n i X j ∈ I p iji ≤ n I µ, where n I = P i ∈ I n i . In particular, for any vertex u and v ∈ X i ( u ) , we have | X i ∗ ( v ) ∩ X I ( u ) | ≤ µn I /n i .Fix a vertex v ∈ V . For some integer ℓ , we fix distinct vertices u , . . . , u ℓ in X i ∗ ( v ) such that for all ≤ α, β ≤ ℓ , we have u α / ∈ X I ( u β ) . Since | X i ∗ ( v ) ∩ X I ( u α ) | ≤ µn I /n i , we may take ℓ = ⌊ n i / (2 µ ) ⌋ . As µ = o ( n i ) by Proposition 7.8,we therefore have ℓ = Ω( n i /µ ) .By Lemma 7.9, for α = β , we have | X I ( u α ) ∩ X I ( u β ) | ≤ µ . Hence, for any ≤ α ≤ ℓ , we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X I ( u α ) \ [ β = α X I ( u β ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & n I − (cid:22) n i µ (cid:23) µ ≥ n I . c ( u α , v ) = i , so v ∈ K I ( u α ) , and so X I ( u α ) \ { v } ⊆ K I ( u α ) ⊆ { v } ⊆ N ( v ) .Then | N ( v ) | ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ℓ [ α =1 X I ( u α , v ) \ { v } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) & n I ℓ n i µ ) = ω ( ρ ) by Proposition 7.8. But this contradicts the definition of ρ . We conclude that λ i < n i − . Lemma 7.15.
Let X be a PCC. Suppose for some nondominant color i we have λ i < n i − . Then λ i ≤ (1 / n i + µ ) .Proof. Fix a vertex u , and suppose λ i < n i − . Then there exist vertices v, w ∈ X i ( u ) such that c ( v, w ) is dominant. Then | N ( v ) ∩ N ( w ) | = µ . Therefore, λ i − µ ≤ | ( N ( u ) ∪ N ( v )) ∩ X i ( u ) | ≤ n i . Corollary 7.16.
Suppose X is a PCC with ρ = o ( n / ) . Then for every nondomi-nant color i , we have λ i . n i / .Proof. For every nondominant color i we have λ i < n i − by Lemma 7.14. Thenby Lemma 7.15 and Proposition 7.8, we have λ i ≤ (1 / n i + µ ) ∼ n i / . Corollary 7.17.
Let X be a PCC with ρ = o ( n / ) with an asymptotically uniformclique geometry C . Then for every nondominant color i there is an integer m i ≥ such that λ i ∼ n i /m i .Proof. Fix a nondominant color i and a vertex u , and let m i be the number ofcliques C ∈ C such that u ∈ C and X i ( u ) ∩ C = ∅ . So n i /m i ∼ λ i . But byCorollary 7.16, we have λ i . n i / , so m i ≥ . In this section we will classify PCCs X having a clique geometry C and a vertexbelonging to at most two cliques of C . In particular, we prove Theorem 2.5.We will assume the hypotheses of Theorem 2.5. So, X will be a PCC suchthat ρ = o ( n / ) , with an asymptotically uniform clique geometry C and a vertex u ∈ V belonging to at most two cliques of C . Lemma 8.1.
Under the hypotheses of Theorem 2.5, for n sufficiently large, everyvertex x ∈ V belongs to exactly two cliques of C , each of order ∼ ρ/ . roof. Recall that by the definition of a clique geometry, for every vertex x ∈ V ,every nondominant color i , and every clique C in the geometry containing x , wehave | C ∩ X i ( x ) | . λ i . Thus, by Corollary 7.16, every vertex belongs to at least twocliques. In particular, u belongs to exactly two cliques of C , and (by Corollary 7.16)it follows that λ i ∼ n i / for every nondominant color i . Hence, by the definitionof a clique geometry, for every vertex x and every nondominant color i , there areexactly two cliques C ∈ C such that x ∈ C and X i ( x ) ∩ C = ∅ .Let i and j be nondominant colors, and let v ∈ X j ( u ) , and let C ∈ C be theclique containing u and v . Since | X i ( u ) ∩ C | ∼ λ i ∼ n i / , we have | N ( v ) ∩ X i ( u ) | & n i / . (7)Now suppose for contradiction that some x ∈ V belongs to at least threecliques of C . Then there is some C ∈ C and nondominant color i such that x ∈ C but X i ( x ) ∩ C = ∅ . Let j be a nondominant color such that X j ( x ) ∩ C = ∅ , and let y ∈ X j ( x ) ∩ C . By the coherence of X and Eq. (7), we have | N ( y ) ∩ X i ( x ) | & n i / .But since there are exactly two cliques C ′ ∈ C such that x ∈ C ′ and X i ( x ) ∩ C ′ = ∅ , then one of these cliques C ′ is such that | N ( y ) ∩ X i ( x ) ∩ C ′ | & n i / . ByProposition 7.8, n i / ω ( µ ) for n sufficiently large. But then C ′ ⊆ N ( y ) , and y / ∈ C ′ , contradicting the maximality of C ′ .So every vertex x ∈ V belongs to exactly two cliques of C , and for each clique C ∈ C containing x and each nondominant color i , we have | X i ( x ) ∩ C | ∼ n i / .It follows that | C | ∼ ρ/ for each C ∈ C . Lemma 8.2.
Under the hypotheses of Theorem 2.5, for n sufficiently large, X hasrank at most four.Proof. Counting the number of vertex–clique incidences in G ( X ) , we have n ∼|C| ( ρ/ ∼ |C| ρ/ by Lemma 8.1. On the other hand, every pair of distinctcliques C, C ′ ∈ C intersects in at most one vertex in G ( X ) by Property 2 of Def-inition 2.3, and so |C| / & n . It follows that ρ . √ n . On the other hand,by Lemma 2.2, we have n i & √ n for every i > . Since ρ = P i> n i , for n sufficiently large there are at most two nondominant colors. Lemma 8.3.
Under the hypotheses of Theorem 2.5, let w be a vertex, and C , C ∈C be the two cliques containing w . Then for any v = w in C , we have | N ( v ) ∩ ( C \ { w } ) | ≤ .Proof. We first note that by Lemma 8.1, there are indeed exactly two cliques con-taining w . Note that v / ∈ C , since otherwise there are two cliques in C containingboth w and v . Suppose v has two distinct neighbors x, y in C \ { w } , so x, y / ∈ C for the same reason. Let C ∈ C \ { C } be the unique clique containing v other39han C . We have x, y ∈ C , but then | C ∩ C | ≥ , a contradiction. So v has atmost one neighbor in C \ { w } .The following result is folklore, although we could not find an explicit state-ment in the literature. A short elementary proof can be found inside the proofof [13, Lemma 4.13]. Lemma 8.4.
Let G be a connected and co-connected strongly regular graph. If G is the line-graph of a graph, then G is isomorphic to T ( m ) , L ( m ) , or C .Proof of Theorem 2.5. Let H be the graph with vertex set C , and an edge { C, C ′ } whenever | C ∩ C ′ | 6 = 0 . Then G ( X ) is isomorphic to the line-graph L ( H ) .By Lemma 8.2, X has rank at most four. By assumption (see Section 3), X hasrank at least three.Consider first the case that X has rank three. The nondiagonal colors i, j of arank three PCC X satisfy either i ∗ = i and j ∗ = j , in which case X is a stronglyregular graph, or i ∗ = j , in which case X is a “strongly regular tournament,” and ρ = ( n − / . We have assumed ρ = o ( n / ) , so X is a strongly regualr graph.But G ( X ) is the line-graph of the graph H , so by Lemma 8.4, for n > , X isisomorphic to either X ( T ( m )) or X ( L ( m )) .Suppose now that X has rank four, and let I = { , } be the nondominantcolors. Fix u ∈ V , and let C , C ∈ C be the cliques containing u by Lemma 8.1.By Corollary 7.16 and Lemma 8.3, for any i, j ∈ I , not necessarily distinct, thereexist v ∈ C and w ∈ C with c ( v, w ) = 1 , c ( v, u ) = i , and c ( u, w ) = j .Therefore, p ij ≥ , and so µ P i,j> p ij ≥ . Now let x ∈ V be such that c ( u, x ) =1 , and let D , D ∈ C be the cliques containing x . For any α, β ∈ { , } , we have | C α ∩ D β | ≤ , and so µ ≤ . Hence, µ = 4 , and | C α ∩ D β | = 1 for every α, β ∈ { , } . Therefore, for any pair of distinct cliques C, C ′ ∈ C we have | C ∩ C ′ | = 1 , and so H is isomorphic to K m , where m = |C| .In particular, every clique C ∈ C has order m − , and so n + n = 2( m − .Now we prove ∗ = 3 and ∗ = 2 . Suppose for contradiction that colors and are symmetric. Fix two vertices u and v with c ( u, v ) = 1 . (See Figure 2.) Then N ( u ) ∩ N ( v ) = { w, x, y, z } for some vertices w, x, y, z ∈ V , and there are fourdistinct cliques C , C , C , C ∈ C such that every vertex in A = { u, v, w, x, y, z } lies in the intersection of two of these cliques. Without loss of generality, assume c ( w, x ) and c ( y, z ) are dominant, and all other distinct pairs in A except ( u, v ) have nondominant color. Since for any i, j ∈ I we have p ij = 1 , then withoutloss of generality, by considering the paths of length two from u to v in G ( X ) , wehave c ( u, w ) = c ( u, x ) = 2 , c ( u, y ) = c ( u, z ) = 3 , c ( v, w ) = c ( v, y ) = 2 , and c ( v, x ) = c ( v, z ) = 3 . Now c ( w, u ) = c ( w, v ) = 2 , and so c ( w, y ) = c ( w, z ) = 3 since p ij = 1 for all i, j ∈ I and c ( w, x ) = 1 . But now c ( u, z ) = c ( v, z ) = vw xyz Figure 2: Two non-adjacent vertices u , v and their common neighbors w, x, y, z .The dashed line represents color 1. The red line represents color 2. The blue linerepresent color 3. c ( w, z ) = 3 , which contradicts the fact that p = p = 1 for c ( z, y ) = 1 . Weconclude that ∗ = 3 and ∗ = 2 .Finally, we prove that individualizing O (log n ) vertices suffices to completelysplit the PCCs of situation (b) of Theorem 2.5. Proof of Lemma 3.5.
By Theorem 2.5, we may assume that X is a rank four PCCwith a non-symmetric nondominant color i , and G ( X ) is isomorphic to T ( m ) for m = n i + 2 . (The other nondominant color is i ∗ .) In particular, every clique in C has order n i + 1 . We show that there is a set of size O (log n ) which completelysplits X .Note that p iii ∗ = p iii = p ii ∗ i by Proposition 5.5. For any edge { u, v } in T ( m ) ,there are exactly m − n i vertices w adjacent to both u and v . Hence, consid-ering all the possible of colorings of these edges in X , we have n i = p iii + p iii ∗ + p ii ∗ i + p ii ∗ i ∗ = 3 p iii + p ii ∗ i ∗ . Therefore, p iii + p ii ∗ i ∗ ≥ n i / , and p iii ∗ + p ii ∗ i ≤ n i / . (8)Fix an arbitrary clique C ∈ C and any pair of distinct vertices u, v ∈ C . (Bypossibly exchanging u and v , we have c ( u, v ) = i .) Of the n i − vertices w wx yij j Figure 3: ( u, w, x, y ) has Property Q ( i, j ) . The dotted line represents the dominantcolor.in C \ { u, v } , at most n i / of these have c ( w, u ) = c ( w, v ) , by Eq. (8). So,including u and v themselves, there are at least n i / − n i / vertices w ∈ C such that c ( w, u ) = c ( w, v ) . Thus, if we individualize a random vertex w ∈ C , then Pr[ c ( w, u ) = c ( w, v )] > / . If this event occurs, then u and v getdifferent colors in the stable refinement. Hence, if we individualize each vertexof C independently at random with probability n i ) /n i , then u and v get thesame color in the stable refinement with probability ≤ /n i . The union boundthen gives a positive probability to every pair of vertices getting a different color,so there is a set A of size O (log n i ) such that after individualizing each vertex in A and refining to the stable coloring, every vertex in C has a uniqe color. We repeatthis process for another clique C ′ , giving every vertex in C ′ a unique color at thecost of another O (log n i ) individualizations.On the other hand, every other clique C ′′ ∈ C intersects C ∪ C ′ in two uniquelydetermined vertices, since G ( X ) is isomorphic to T ( m ) . So, if u ∈ C ′′ and v / ∈ C ′′ ,then u and v get different colors in the stable refinement. Since every vertex lies intwo uniquely determined cliques by Lemma 8.1, it follows that every vertex gets adifferent color in the stable refinement. In this section, we finally prove Lemma 3.4. For given nondominant colors i and j , we will be interested in quadruples of vertices ( u, w, x, y ) with the followingproperty: Property Q ( i, j ) : c ( x, y ) = c ( u, x ) = c ( u, y ) = 1 , c ( u, w ) = i , and c ( w, x ) = c ( w, y ) = j (See Figure 3) 42 efinition 9.1 (Good triple of vertices) . For fixed nondominant colors i, j andvertices u, v , we say a triple of vertices ( w, x, y ) is good for u and v if ( u, w, x, y ) has Property Q ( i, j ) , but there is no vertex z such that ( v, z, x, y ) has Property Q ( i, j ) .We observe that if ( w, x, y ) is good for vertices u and v , and both x and y areindividualized, then u and v receive different colors after two refinement steps.In the case of SRGs, there is only one choice of nondominant color, and Prop-erty Q ( i, j ) and Definition 9.1 can be simplified: a triple ( w, x, y ) is good for u and v if w, x, y, u induces a K , , but there is no vertez z such that z, x, y, v in-duces a K , . Careful counting of induced K , subgraphs formed a major partof Spielman’s proof of Theorem 1.4 in the special case of SRGs [25]. Spielman’sideas inspired parts of this section. In particular, the proof of the following lemmadirectly generalizes Lemmas 14 and 15 of [25]. Lemma 9.2.
Let X be a PCC with ρ = o ( n / ) . Suppose that for every distinct u, v ∈ V there are nondominant colors i and j such that there are α = Ω( n i n j ) good triples ( w, x, y ) of vertices for ( u, v ) . Then there is a set of O ( n / (log n ) / ) vertices that completely splits X .Proof. Let S be a random set of vertices given by including each vertex in V in-dependently with probability p . Fix distinct u, v ∈ V . We estimate the probabilitythat there is a good triple ( w, x, y ) for u and v such that x, y ∈ S .Let T denote the set of good triples ( w, x, y ) of vertices for ( u, v ) . Observethat any vertex w ∈ X i ( u ) appears in at most n j good triples ( w, x, y ) in T . Onthe other hand, if w ∈ X i ( u ) is a random vertex, and X is the number of pairs x, y such that ( w, x, y ) ∈ T , then E [ X ] ≥ α/n i . Therefore, we have n j Pr[ X ≥ α/ (2 n i )] + (1 − Pr[ X ≥ α/ (2 n i )]) α/ (2 n i ) ≥ E [ X ] ≥ α/n i , and so, since α = Ω( n i n j ) and α < n i n j by definition, Pr[ X ≥ α/ (2 n i )] ≥ n j n i /α − . Let U be the set of vertices w ∈ X i ( u ) appearing in at least α/ (2 n i ) triples ( w, x, y ) in T , so | U | = Ω( n i ) . Now let W ⊆ U be a random set given by in-cluding each vertex w ∈ U independently with probability n/ (3 n i n j ) .Fix a vertex w ∈ W and a triple ( w, x, y ) ∈ T . Note that there are at most p ij . n i n j /n vertices w ′ ∈ U such that c ( w ′ , x ) = j . Therefore, by the unionbound, the probability that there is some w ′ = w with w ′ ∈ X j ∗ ( x ) ∩ W is ≤ / .43imilarly, the probability that there is some w ′ ∈ X j ∗ ( y ) ∩ W with w ′ = w is atmost / . Hence, the probability that X j ∗ ( x ) ∩ W = X j ∗ ( y ) ∩ W = { w } (9)is at least / .Now for any w ∈ W , let T w denote the set of pairs x, y ∈ V such that ( w, x, y ) ∈ T and Eq. (9) holds. We have E [ | T w | ] ≥ α/ (6 n i ) = Ω( n j ) . Butin any case, | T w | ≤ n j . Therefore, for any w ∈ W , we have | T w | = Ω( n j ) withprobability Ω(1) . Let W ′ ⊆ W be the set of vertices w with | T w | = Ω( n j ) . Since E [ | W ′ | ] = Ω( | W | ) , we have | W ′ | = Ω( | W | ) with probability Ω(1) . Furthermore, | W | = Ω( n/n j ) with high probability by the Chernoff bound.Thus, there exists a set W ⊆ X i ( x ) with a subset W ′ ⊆ W of size Ω( n/n j ) such that | T w | = Ω( n j ) for every w ∈ W ′ .Now fix a w ∈ W ′ . The probability that there are at least two vertices in X j ( w ) ∩ S is at least − (1 − p ) n j − pn j (1 − p ) n j − > − e − pn j − pn j e − pn j = Ω( p n j ) if pn j < , using the Taylor expansion of the exponential function. Since | T w | =Ω( n j ) , the probability that there is a pair ( x, y ) ∈ T w with x, y ∈ S is Ω( p n j ) .Therefore, the probability that there is no w ∈ W ′ with a pair ( x, y ) ∈ T w suchthat x, y ∈ S is at most (1 − Ω( p n j )) | W ′ | ≤ (1 − Ω( p n j )) εn/n j , for some constant < ε < . For p = β p log n/ ( nn j ) with a sufficientlylarge constant β , this probability is at most / (2 n ) . Since n j & √ n for all j by Lemma 2.2, we may take p = β p log n/n / with a sufficiently large con-stant β . Then, for any pair u, v ∈ V of distinct vertices, the probability no goodtriple ( w, x, y ) for u and v has x, y ∈ S is at most / (2 n ) . By the union bound,the probability that there is some pair u, v ∈ V of distinct vertices such that notriple ( w, x, y ) has the desired property is at most / . Therefore, after indi-vidualizing every vertex in S , every vertex in V gets a unique color with prob-ability at least / . By the Chernoff bound, we may furthermore assume that | S | = O ( n / (log n ) / ) .We will prove that the hypotheses of Lemma 9.2 hold separately for the casethat λ k is small for some nondominant color k and the case that X has an asymptot-ically uniform clique geometry. Specifically, in Section 9.1 we prove the followinglemma. 44 emma 9.3. There is an absolute constant ε > such that the following holds. Let X be a PCC with ρ = o ( n / ) and a nondominant color k such that λ k < εn / .Then there are two nondominant colors i and j such that for every pair of distinctvertices u, v ∈ V , there are Ω( n i n j ) good triples of vertices for u and v withrespect to the colors i and j . Then, in Section 9.2, we prove the following lemma.
Lemma 9.4.
Let X be a PCC with ρ = o ( n / ) and a asymptotically uniformclique geometry C such that every vertex u ∈ V belongs to at least three cliquesin C . Suppose that n i µ = o ( λ i ) for every nondominant color i . Then there arenondominant colors i and j such that for every pair of distinct vertices u, v ∈ V ,there are Ω( n i n j ) good triples of vertices for u and v with respect to the colors i and j . Lemma 3.4 follows from Lemmas 9.2, 9.3, and Lemma 9.4 (noting for the latterthat λ i = Ω( n / ) implies n i µ = o ( λ i ) by Proposition 7.8).Before proving Lemmas 9.3 and 9.4, we prove two smaller lemmas that will beuseful for both. Lemma 9.5.
Let X be a PCC with ρ = o ( n / ) . Let i be a nondominant color, andlet u, v ∈ V be distinct vertices. We have | X i ( u ) \ N ( v ) | & n i / .Proof. Let j = c ( u, v ) , and let ε = 2 µ/n j , so ε = o (1) by Proposition 7.8. ByCorollary 7.16, we have λ i . n i / . Therefore, by Lemma 7.11, we have | X i ( u ) ∩ N ( v ) | ≤ max (cid:26) λ i + 11 − ε , n i µεn j (cid:27) . n i / . Lemma 9.6.
Let X be a PCC with ρ = o ( n / ) . Let i and j be nondominant colorsand let u and w be vertices with c ( u, w ) = i . Suppose that | X j ( w ) ∩ N ( u ) | . n j / . Then there are & (1 / n j pairs of vertices ( x, y ) with c ( x, y ) = 1 such that ( u, w, x, y ) has Property Q ( i, j ) .Proof. By Corollary 7.16, we have λ j . n j / . Thus, for every vertex x ∈ X j ( w ) \ N ( u ) , there are at least n j − | X j ( w ) ∩ N ( u ) | − λ j & n j / vertices y ∈ X j ( w ) \ N ( u ) such that ( u, w, x, y ) has Property Q ( i, j ) . Since | X j ( w ) ∩ N ( u ) | . n j / ,the number of pairs ( x, y ) with c ( x, y ) = 1 such that ( u, w, x, y ) has Property Q ( i, j ) is & (2 n j / n j /
6) = (1 / n j .45 .1 Good triples when some parameter λ k is small We prove Lemma 9.3 in two parts, via the following two lemmas.
Lemma 9.7.
There is an absolute constant ε > such that the following holds. Let X be a PCC with ρ = o ( n / ) and a nondominant color k such that λ k < εn / and λ k ∗ . n k / . Then there are two nondominant colors i and j such that forevery pair of distinct vertices u, v ∈ V , there are Ω( n i n j ) good triples of verticesfor u and v with respect to the colors i and j . Lemma 9.8.
Let τ be an arbitrary fixed positive integer. Let X be a PCC with ρ = o ( n / ) and a nondominant color k such that λ k < n / / ( τ + 1) and λ k ∗ & n k /τ .Then for every pair of distinct vertices u, v ∈ V , there are Ω( n k ) good triples ofvertices for u and v with respect to the colors i = k ∗ and j = k ∗ . We observe that Lemma 9.3 follows from these two.
Proof of Lemma 9.3.
Let ε ′ be the absolute constant given by Lemma 9.7, and let ε = min { ε ′ , / } . Let X be a PCC with ρ = o ( n / ) and a nondominant color k such that λ k < εn / . If λ k ∗ & n k / , then Lemma 9.8 gives the desired result.Otherwise, the result follows from Lemma 9.7.We now turn our attention to proving Lemmas 9.7. Lemma 9.9.
Let < ε < be a constant, and X be a PCC with ρ = o ( n / ) .Let i and k be nondominant colors, and let w and v be vertices such that c ( w, v ) is dominant. Suppose n i ≤ n ℓ for all ℓ , and λ k < εn / . Then there are . ε n k triples ( z, x, y ) of vertices such that x, y ∈ X k ∗ ( w ) , c ( x, y ) = 1 and ( v, z, x, y ) has Property Q ( i, k ∗ ) .Proof. First we observe that ( p k ∗ k ) n i . ( n k /n ) n i ≤ n k ( ρ /n ) = o ( n k ) . For every color ℓ , there are exactly p ℓi ∗ vertices z such that c ( v, z ) = i and c ( w, z ) = ℓ . For every such vertex z , there are at most ( p ℓk ∗ k ) pairs x, y ∈ X k ∗ ( w ) with c ( x, y ) = 1 such that ( v, z, x, y ) has Property Q ( i, k ∗ ) . Thus, byProposition 5.5, the total number of such triples is r − X ℓ =1 p ℓi ∗ ( p ℓk ∗ k ) . p i ∗ ( p k ∗ k ) + r − X ℓ =2 n ℓ n i n (cid:18) n k p k ∗ ℓk ∗ n ℓ (cid:19) ≤ o ( n k ) + r − X ℓ =2 (cid:18) n k n (cid:19) (cid:16) p kkℓ ∗ (cid:17) ≤ o ( n k ) + n k n λ k . ε n k .
46e finally complete the proof of Lemma 9.7.
Proof of Lemma 9.7.
Let ε = p / . Let i be a nondominant color minimizing n i . By Lemma 9.5, we have | X i ( u ) \ N ( v ) | & n i / for any pair of distinct vertices u, v ∈ V .Let color j = k ∗ . Fix two distinct vertices u and v . Let w ∈ X i ( u ) \ N ( v ) .Since λ j . n j / , we have | X j ( w ) ∩ N ( u ) | . n j / by Lemma 7.11 (with ε = p µ/n i = o (1) ). By Lemma 9.6, there are & (1 / n j pairs of vertices ( x, y ) with c ( x, y ) = 1 such that ( u, w, x, y ) has Property Q ( i, j ) . Furthermore, byLemma 9.9, for all but . (1 / n j of these pairs ( x, y ) , the triple ( w, x, y ) isgood for u and v . Since there are & n i / such vertices w , we have a total of & (1 / n i n j triples ( w, x, y ) that are good for u and v .Now we prove Lemma 9.8. Lemma 9.10.
Let X be a PCC with ρ = o ( n / ) and strong I -local clique par-titions for some set I of nondominant colors. Let j ∈ I be a color such that λ j = Ω( n j ) . Let w and v be vertices such that c ( w, v ) = 1 . Then for any nondom-inant color i with n i ≤ n j , there are o ( n j ) triples ( z, x, y ) of vertices such that x, y ∈ X j ( w ) , c ( x, y ) = 1 and ( v, z, x, y ) has Property Q ( i, j ) .Proof. Fix a nondominant color i , and let T be the set of triples ( z, x, y ) such that x, y ∈ X j ( w ) , c ( x, y ) = 1 and ( v, z, x, y ) has Property Q ( i, j ) .If c ( z, w ) = 1 , then | X j ( z ) ∩ X j ( w ) | = p jj ∗ , and so there are at most ( p jj ∗ ) pairs x, y ∈ X j ( w ) with c ( x, y ) = 1 such that ( z, x, y ) ∈ T . Then since c ( v, z ) = i whenever ( z, x, y ) ∈ T , the total number of triples ( z, x, y ) ∈ T such that C ( z, w ) = 1 is at most ( p jj ∗ ) n i . ( n j /n ) n i ≤ n j ( ρ /n ) = o ( n j ) , where the first inequality follows from Proposition 5.5 (iii) and (iv), and the relation n ∼ n .Since c ( w, v ) = 1 , there are ≤ ρn i /n vertices z ∈ X i ( v ) ∩ N ( w ) . Suppose z ∈ X i ( v ) ∩ N ( w ) . Let C denote the collection of cliques partitioning X I ( w ) . If someclique in C contains z , let C be that clique; otherwise, let C = ∅ . Since C partitions X j ( w ) into ∼ n j /λ j = O (1) cliques for each w ∈ V , and | X j ( z ) ∩ C ′ | ≤ µ forevery clique C ′ ∈ C with C = C ′ , we therefore have | ( X j ( w ) ∩ X j ( z )) \ C | . µn j /λ j = O ( µ ) . But then there are at most | X j ( w ) ∩ X j ( z ) | · | ( X j ( w ) ∩ X j ( z )) \ C | = O ( n j µ ) x, y ∈ X j ( w ) with c ( x, y ) = 1 such that ( v, z, x, y ) ∈ T , for a total of atmost O ( n j µ ( ρn i ) /n ) = o ( n j ) triples ( z, x, y ) ∈ T with c ( z, w ) = 1 . Proof of Lemma 9.8.
For every nondominant color ℓ , by Proposition 7.8, we have n k p µ/n ℓ = o ( n k ) = o ( λ k ∗ ) . Similarly, n k µ = o ( λ k ∗ ) . Hence, by Lemma 7.13and Definition 7.6, there is a set I of nondominant colors with k ∗ ∈ I such that X has strong I -local clique partitions. Since λ k < n / / ( τ + 1) . n k / ( τ + 1) by Lemma 2.2, and since λ k ∗ & n k /τ , the by the definition of an I -local cliquepartition, k / ∈ I . Hence, by the definition of a strong I -local clique partition andObservation 7.4, for a vertex x and a vertex y ∈ X k ( x ) , | N ( y ) ∩ X k ∗ ( x ) | ≤ τ µ = o ( n k ) .On the other hand, by Corollary 7.16, we have λ k ∗ . n k / , and hence λ k ≤ n / / . n k / by Lemma 2.2.Fix u, v ∈ V . By Lemma 9.5, we have | X k ∗ ( u ) \ N ( v ) | & n k ∗ / . Let w ∈ X k ∗ ( u ) \ N ( v ) . We have c ( w, u ) = k and so | X k ∗ ( w ) ∩ N ( u ) | = o ( n k ) .By Lemma 9.6, there are Ω( n k ) pairs of non-adjacent vertices ( x, y ) such that ( u, w, x, y ) has Property Q ( k ∗ , k ∗ ) . But by Lemma 9.10, there are o ( n k ) triples ( x, y, z ) of vertices such that x, y ∈ X k ∗ ( w ) , c ( x, y ) = 1 and ( v, z, x, y ) has Prop-erty Q ( k ∗ , k ∗ ) . So there are Ω( n k ) pairs ( x, y ) of vertices such that ( w, x, y ) isgood for u and v with respect to colors i = k ∗ and j = k ∗ . Since we have Ω( n k ) choices for vertex w , there are in total Ω( n k ) good triples, as desired. We now prove Lemma 9.4.
Lemma 9.11.
Let X be a PCC with ρ = o ( n / ) and an asymptotically uniformclique geometry C such that every vertex u ∈ V belongs to at least three cliquesin C . Suppose that n i µ = o ( λ i ) for every nondominant color i . Then, for anynondominant color i , there is a nondominant color j such that for every u, w with w ∈ X i ( u ) , we have | X j ( w ) ∩ N ( u ) | . n j / .Proof. Let C ∈ C be the unique clique such that u, w ∈ C .If λ i ∗ . n i / , we let j = i ∗ . Then by the maximality of the cliques in C partitioning X j ( w ) , we have | ( X j ( w ) ∩ N ( u )) \ C | ≤ µn j /λ j = o ( λ j ) by Observation 7.4. So | X j ( w ) ∩ N ( u ) | . n j / .Otherwise, by Corollary 7.17, λ i ∗ ∼ n i / , and so there is at most one clique C ′ ∈ C with C ′ = C such that | X i ∗ ( w ) ∩ C ′ | 6 = 0 . Therefore, there is a clique C ′′ such that | X i ∗ ( w ) ∩ C ′′ | = 0 . Let j be a nondominant color such that | X j ( w ) ∩ ′′ | ∼ λ j . Again, by the maximality of the cliques in C partitioning X j ( w ) andObservation 7.4, we have | X j ( w ) ∩ N ( u ) | . µn j /λ j = o ( λ j ) , as desired. Furthermore, this inequality does not depend on the choice of w by thecoherence of X . Lemma 9.12.
Let X be a PCC with ρ = o ( n / ) and an asymptotically uniformclique geometry C , let w and v be vertices such that c ( w, v ) is dominant, andlet j be a nondominant color such that µ = o (min { λ j , λ j /n j } ) . Then for anynondominant color i , there are o ( n j ) triples ( z, x, y ) of vertices such that x, y ∈ X j ( w ) , c ( x, y ) = 1 and ( v, z, x, y ) has Property Q ( i, j ) .Proof. Fix a nondominant color i , and let T be the set of triples ( z, x, y ) such that x, y ∈ X j ( w ) , c ( x, y ) = 1 and ( v, z, x, y ) has Property Q ( i, j ) .If c ( z, w ) = 1 , then | X j ( z ) ∩ X j ( w ) | = p jj ∗ , and so there are at most ( p jj ∗ ) pairs x, y ∈ X j ( w ) such that ( z, x, y ) ∈ T , for a total of at most ( p jj ∗ ) n i . ( n j /n ) n i ≤ n j ( ρ /n ) = o ( n j ) . triples ( z, x, y ) ∈ T with c ( z, w ) = 1 .Since c ( w, v ) = 1 , there are ≤ µ vertices z ∈ X i ( v ) ∩ N ( w ) . Suppose z ∈ X i ( v ) ∩ N ( w ) . Let C be the clique in C containing both z and w . Note that | C ∩ X j ( w ) | . λ j . For any C w , C z ∈ C with w ∈ C w and z ∈ C z such that C w = C and C z = C , we have | C w ∩ C z | ≤ . Since C partitions X j ( u ) into ∼ n j /λ j cliques for each u ∈ V , we therefore have | ( X j ( w ) ∩ X j ( z )) \ C | . ( n j /λ j ) . Butthen there are at most | X j ( w ) ∩ X j ( z ) | · | ( X j ( w ) ∩ X j ( z )) \ C | . ( λ j + ( n j /λ j ) )( n j /λ j ) . n j /λ j + ( n j /λ j ) = o ( n j /µ ) pairs x, y ∈ X j ( w ) with c ( x, y ) = 1 such that ( v, z, x, y ) ∈ T , for a total of atmost o ( n j ) triples ( z, x, y ) ∈ T with c ( z, w ) = 1 . Proof of Lemma 9.4.
Let u and v be two distinct vertices. By Lemma 9.5, there isa nondominant color i such that | X i ( u ) \ N ( v ) | & n i / . By Lemma 9.11, there isa nondominant color j such that for every w ∈ X i ( u ) , we have | X j ( w ) ∩ N ( u ) | . n j / .Let w ∈ X i ( u ) \ N ( v ) . By Lemma 9.6, there are Ω( n j ) pairs of vertices ( x, y ) with c ( x, y ) = 1 such that ( u, w, x, y ) has Property Q ( i, j ) . Furthermore, since µ is a positive integer and µ = o ( λ j /n j ) , we have µ = o (min { λ j , λ j /n j } ) . By49emma 9.12, for all but o ( n j ) of these pairs ( x, y ) , the triple ( w, x, y ) is good for u and v . Since there are & n i / such vertices w , we have a total of Ω( n i n j ) triples ( w, x, y ) that are good for u and v .
10 Conclusion
We have proved that except for the readily identified exceptions of complete, trian-gular, and lattice graphs, a PCC is completely split after individualizing e O ( n / ) vertices and applying naive color refinement. Hence, with only those three classesof exceptions, PCCs have at most exp( e O ( n / )) automorphisms. As a corollary,we have given a CFSG-free classifcation of the primitive permutation groups ofsufficiently large degree n and order not less than exp( e O ( n / )) .As we remarked in the introduction, Theorem 1.4 is tight up to polylogarith-mic factors in the exponent, as evidenced by the Johnson and Hamming schemes.However, further progress may be possible for Babai’s conjectured classificationof PCCs with large automorphism groups, Conjecture 1.2.The PCCs with large automorphism groups appearing in Conjecture 1.2 are allin fact association schemes, i.e., they satisfy i ∗ = i for every color i . Intuitively,the presence of asymmetric colors (oriented constituent graphs) should reduce thenumber of automorphisms. On the other hand, the possibility of asymmetric col-ors greatly complicates our analysis. For example, situation (2) of Theorem 2.5and Lemma 3.5 could be eliminated, and the proof of Lemma 2.2 would becomestraightforward, for association schemes. Hence, a reduction to the case of associ-ation schemes would be desirable. Question 1.
Is it the case that every sufficiently large PCC with at least exp( n ε ) automorphisms is an association scheme?The best that is known in this direction is the result of the present paper: if X is a PCC that is not an association scheme, then | Aut( X ) | ≤ exp( e O ( n / )) .We comment on the bottlenecks for the current analysis. Below the threshold ρ = o ( n / ) , we in fact have the improved bound | Aut( X ) | ≤ exp( e O ( n / )) when X is nonexceptional, by Lemma 3.4. This region of the parameters is therefore nota bottleneck for improving the current analysis. On the other hand, Conjecture 1.2suggests that nonexceptional PCCs X with ρ = o ( n / ) should satisfy | Aut( X ) | ≤ exp( O ( n o (1) )) .When ρ = Θ( n / ) , the Johnson scheme J ( m, and H (3 , m ) emerge as ad-ditional exceptions, with automorphism groups of order exp(Θ( n / log n )) . Thebottleneck for the current analysis is above this threshold. In this region of the pa-rameters, we analyze the distinguishing number D ( i ) of the edge-colors i . When50 < n/ , our best bounds are D (1) = Ω( ρ ) and D ( i ) ≥ Ω( D (1) n/ρ ) / fromLemma 5.13. When ρ ≥ n/ , we use the estimate D ( i ) ≥ Ω( p ρn i / log n ) fromLemma 5.14. Neither bound simplifies to anything better than D ( i ) = Ω( n / ) inany portion of the range range ρ = Ω( n / ) .Babai makes the following conjecture [1, Conjecture 7.4], which would givean improvement when ρ ≥ n / ε . Conjecture 10.1.
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