Hypergraphs with many extremal configurations
HHYPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS
XIZHI LIU, DHRUV MUBAYI, AND CHRISTIAN REIHER
Abstract.
For every positive integer t we construct a finite family of triple systems M t ,determine its Turán number, and show that there are t extremal M t -free configurationsthat are far from each other in edit-distance. We also prove a strong stability theorem:every M t -free triple system whose size is close to the maximum size is a subgraph of oneof these t extremal configurations after removing a small proportion of vertices. This isthe first stability theorem for a hypergraph problem with an arbitrary (finite) number ofextremal configurations. Moreover, the extremal hypergraphs have very different shadowsizes (unlike the case of the famous Turán tetrahedron conjecture). Hence a corollary ofour main result is that the boundary of the feasible region of M t has exactly t globalmaxima. § Introduction
Stability.
Let r ě F be a family of r -uniform hypergraphs (henceforthcalled r -graphs). An r -graph H is F -free if it contains no member of F as a subgraph.For every natural number n the Turán number ex p n, F q of F is the maximum numberof edges in an F -free r -graph on n vertices. The Turán density π p F q of F is definedas π p F q “ lim n Ñ8 ex p n, F q{ ` nr ˘ , and F is nondegenerate if π p F q ą
0. By a theorem ofErdős [ ], this is equivalent to F containing an r -graph which is not r -partite.The study of ex p n, F q is perhaps the central topic in extremal graph and hypergraph the-ory. Curiously, unlike the case for graphs, determining π p F q for a family F of hypergraphsis known to be notoriously hard in general. Indeed, the problem of determining π p K r‘ q raised by Turán [ ], where K r‘ is the complete r -graph on ‘ vertices, is still wide open forall ‘ ą r ě
3. Erdős offered $500 for the determination of any π p K r‘ q with ‘ ą r ě π p K r‘ q with ‘ ą r ě ] motivated the second author [ ]to make the following definition. A family F of r -graphs is t - stable if for every m P N there exist r -graphs G p m q , . . . , G t p m q on m vertices such that the following holds. Forevery δ ą ε ą n such that for all n ě n if H is an F -free r -graph Key words and phrases. hypergraph Turán problems, stability, feasible regions.The first and second author’s research is partially supported by NSF awards DMS-1763317 and DMS-1952767. a r X i v : . [ m a t h . C O ] F e b XIZHI LIU, DHRUV MUBAYI, AND CHRISTIAN REIHER on n vertices with | H | ą p ´ ε q ex p n, F q , then H can be transformed to some G i p n q by adding and removing at most δ | H | edges.Say F is stable if it is 1-stable. Denote by ξ p F q the minimum integer t such that F is t -stable, and set ξ p F q “ 8 if there is no such t . Call ξ p F q the stability number of F .The Erdős-Stone-Simonovits theorem [ , ] and Erdős-Simonovits stability theorem [ ]imply that every nondegenerate family of graphs is stable. However, for hypergraphs thereare many families (whose Turán densities are unknown) which are conjecturally not stable.Two famous examples are Turán’s conjecture on tetrahedra (e.g. see [ , , ]) and theErdős-Sós conjecture on triple systems with bipartite links (e.g. see [ , ]). In fact, noTurán density of a nondegenerate family of hypergraphs without the stability property wasknown (e.g. see [ ]) until very recently, when the first two authors constructed a 2-stablefamily M of triple systems [ ]. Our first main result states that, more generally, for everynatural number t there exists a family of triple systems satisfying ξ p M t q “ t .We identify an r -graph H with its edge set, use V p H q to denote its vertex set, anddenote by v p H q the size of V p H q . An r -graph H is a blow-up of an r -graph G if thereexists a map ψ : V p H q Ñ V p G q so that ψ p E q P G iff E P H , and we say H is G -colorable ifthere exists a map ϕ : V p H q Ñ V p G q so that ϕ p E q P G for all E P H . In other words, H is G -colorable if and only if H occurs as a subgraph in some blow-up of G . Theorem 1.1.
For every positive integer t there exist constants ă n ă ¨ ¨ ¨ ă n t , ă λ t ă { , t triple systems G , . . . , G t with v p G i q “ n i for i P r t s , and a finite family M t of triple systems with the following properties. ( a ) The inequality ex p n, M t q ď λ t n holds for all positive integers n , and moreover,equality holds whenever n is a multiple of n i for some i P r t s . ( b ) For every δ ą there exist ε ą and N so that the following holds for all n ě N .Every M t -free triple system H on n vertices with at least p λ t ´ ε q n edges can be made G i -colorable for some i P r t s by removing at most δn vertices. Moreover, ξ p M t q “ t . Feasible regions.
Recall that the shadow of an r -graph H is defined to be the p r ´ q -graph B H “ " A P ˆ V p H q r ´ ˙ : there is B P H such that A Ď B * . Call d p H q “ | H |{ ` v p H q r ˘ the edge density and d pB H q “ |B H |{ ` v p H q r ´ ˘ the shadow density of H .Given a family F the feasible region Ω p F q of F is the set of points p x, y q P r , s such that there exists a sequence of F -free r -graphs p H k q k “ with lim k Ñ8 v p H k q “ 8 , YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 3 lim k Ñ8 d pB H k q “ x , and lim k Ñ8 d p H k q “ y . The feasible region unifies and generalizesmany classical problems such as the Kruskal-Katona theorem [ , ] and the Turán problem.It was introduced recently in [ ] to understand the extremal properties of F -free hyper-graphs beyond just the determination of π p F q . The general shape of Ω p F q was analyzedin [ ] as follows: For some constant c p F q P r , s the projection to the first coordinate,projΩ p F q “ t x : there is y P r , s such that p x, y q P Ω p F qu , is the interval r , c p F qs . Moreover, there is a left-continuous almost everywhere differen-tiable function g p F q : projΩ p F q Ñ r , s such thatΩ p F q “ (cid:32) p x, y q P r , c p F qs ˆ r , s : 0 ď y ď g p F qp x q ( . Let us call g p F q the feasible region function of F . There are examples showing that g p F q is not necessarily continuous (see [ , Theorem 1.12]) and the present work is part of aneffort to figure out how “exotic” these functions can be.The stability number of F can give information about the shape of Ω p F q , more precisely,about the number of global maxima of g p F q (e.g. see Proposition ). The family M oftriple systems from [ ] for which ξ p M q “ M far from each other in edit-distance,but the same is true of their shadows. As a consequence, in addition to ξ p M q “
2, thefunction g p M q has exactly two global maxima. The authors raised the question of whetherthere exists a finite family M t of triple systems so that the function g p M t q has exactly t global maxima for t ě , Problem 6.10]). Our second main result asserts that theobjects constructed in the course of proving Theorem give a positive solution to thisproblem. Theorem 1.2.
For every positive integer t there exist constants ă n ă ¨ ¨ ¨ ă n t , ă λ t ă { , and a finite family M t of triple systems such that projΩ p M t q “ r , s , and g p M t , x q ď λ t for all x P r , s . Moreover, g p M t , x q “ λ t if and only if x “ ´ { n i forsome i P r t s . Roughly speaking, the connection between these results is as follows. An r -graph is a star if there is a vertex v such that all edges contain v , and an r -graph H is semibipartite if it is S -colorable for some star S . Note that this is the same as saying that V p H q has a partitioninto two parts A and B such that all edges have exactly one vertex in A and r ´ B . We will see later that our definition of M t ensures that every semibipartite 3-graphis M t -free. By shrinking A , the shadow density of an n -vertex semibipartite 3-graph H canbe made arbitrarily close to 1 as n Ñ 8 , so projΩ p M t q “ r , s . The shadows of the triplesystems G , . . . , G t from Theorem are complete graphs and thus their edge densities are XIZHI LIU, DHRUV MUBAYI, AND CHRISTIAN REIHER10 y x λ t n ´ n n ´ n ¨ ¨ ¨ n t ´ n t p , q Figure 1.1.
The function g p M t q has exactly t global maxima.the distinct numbers 1 ´ { n , . . . , ´ { n t . So g p M t , x q “ λ t holds if x is one of thosedensities and stability allows us to exclude further solutions to this equation. Organization.
In Section we present some definitions related to the Lagrangian ofhypergraphs and prove a result about the Lagrangian of a class of almost complete 3-graphs. In Section we use the result from Section to define the extremal configurations,which are balanced blow-ups of G , . . . , G t , define the forbidden family M t , and prove thefirst part of Theorem . We prove the second part of Theorem in Section , andTheorem in Section . Section contains some concluding remarks on generalisationsto r -graphs and open problems. § Lagrangian
In this section we present some definitions related to the Lagrangian of a hypergraph,introduced by Frankl and Rödl in [ ], and prove a result (Proposition below) aboutcertain almost complete triple systems.Let G be an r -graph for some r ě
2. The neighborhood of a vertex v P V p G q is definedto be N G p v q “ t u P V p G q (cid:114) t v u : there is A P G such that t u, v u Ď A u , the link of v is L G p v q “ t A P B G : A Y t v u P G u , and d G p v q “ | L G p v q| is called the degree of v . Denote by δ p G q , ∆ p G q the minimum andmaximum degree of G , respectively. For a pair of vertices u, v P V p G q the neighborhood of YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 5 t u, v u is N G p u, v q “ t w P V p G q (cid:114) t u, v u : D A P G such that t u, v, w u Ď A u , and d G p u, v q “ | N H p u, v q| is called the codegree of t u, v u . Denote by δ p G q , ∆ p G q theminimum and maximum codegree of G , respectively.For an r -graph G on n vertices (let us assume for notational transparency that V p G q “ r n s )the multilinear function L G : R n Ñ R is defined by L G p x , . . . , x n q “ ÿ E P H ź i P E x i for all p x , . . . , x n q P R n . Denote by ∆ n ´ the standard p n ´ q -dimensional simplex, i.e.∆ n ´ “ tp x , . . . , x n q P r , s n : x ` ¨ ¨ ¨ ` x n “ u . Since ∆ n ´ is compact, a theorem of Weierstraß implies that the restriction of L G to ∆ n ´ attains a maximum value, called the Lagrangian of G and denoted by λ p G q .For a hypergraph G the maximum number of edges in a blow-up of G is related to λ p G q (e.g. see Frankl and Füredi [ ] or Keevash’s survey [ , Section 3]). Lemma 2.1 ([ , ]) . Let r ě and let G , H be two r -graphs. If H is a blow up of G , then | H | ď λ p G q v p H q r . Given a 3-graph G , by plugging p { n, . . . , { n q into L G one immediately obtains the lowerbound λ p L G q ě | G |{ n . It is well known that for cliques H “ K n this holds with equalityand, moreover, that p { n, . . . , { n q is the only point in the simplex ∆ n ´ , where L H attainsthis maximum value.The main result of this section, Proposition below, exhibits a class of almost complete3-graphs having the same properties. This will allow us later to construct for every givenpositive integer t a family t G , . . . , G t u of 3-graphs and a rational number λ t close to 1 { λ p G i q “ | G i |{ v p G i q “ λ t holds for all i P r t s . The extremal configurations for ourhypergraph Turán problem are then going to be balanced blow-ups of G , . . . , G t . As wecan accomplish v p G q ă ¨ ¨ ¨ ă v p G t q , this is relevant to Theorem .Let us observe that every hypergraph G satisfying λ p G q “ | G |{ v p G q needs to be regularin the sense that all vertices have the same degree. In the converse direction, regularhypergraphs can still have much larger Lagrangians than | G |{ v p G q . For instance, theLagrangian of the Fano plane is 1 {
27 but not 1 {
49. To avoid such situations we utilize adesign theoretic construction.For the purposes of this article, by an p n, k q -design we shall mean a k -graph D on n vertices such that every pair of vertices is covered by a unique edge. With every such XIZHI LIU, DHRUV MUBAYI, AND CHRISTIAN REIHER design D we associate the 3-graph H p D q “ ď E P D ˆ E ˙ . on V p D q . Note that | H p D q| “ ˆ k ˙ ` n ˘` k ˘ “ k ´ n p n ´ q . It will turn out that for n ě k every 3-graph of the form G “ K n (cid:114) H p D q , where D isan p n, k q -design on r n s , has the property λ p G q “ | G |{ v p G q . In order to increase our controlover the resulting value of λ p G q Proposition allows the extra flexibility to subtract avery sparse regular 3-graph from G . Moreover, for reasons related to stability we stateslightly more than just the actual value of the Lagrangian. Proposition 2.2.
Suppose that n ě k ` s , D is an p n, k q -design on r n s , and S is an s -regular -graph on r n s . If S X H p D q “ ∅ and G “ K n (cid:114) p H p D q Y S q , then L G p x , . . . , x n q ` n ÿ i “ ˆ x i ´ n ˙ ď | G | n “ ˆ ´ k ` n ` k ´ sn ˙ (2.1) holds for all p x , . . . , x n q P ∆ n ´ and, consequently, λ p G q “ ˆ ´ k ` n ` k ´ sn ˙ . (2.2)We start with a simple observation that will come in handily later. Fact 2.3.
Let G be a -graph with vertex set r n s and let α ě be a real number. If thereal numbers α , . . . , α n P r´ , α s sum up to zero, then L G p α , . . . , α n q ď p αn q . Proof of Fact . Define P “ t i P r n s : α i ą u to be the set of vertices of G with positiveweight. Let us decompose L G p α , . . . , α n q “ S ` S ` S ` S such that for m P t , , , u the sum S m consists of all terms α i α j α k contributing to L G and satisfying | P Xt i, j, k u| “ m .As the sums S and S possess no positive terms, we have S , S ď
0. Moreover, S has nomore than ` | P | ˘ ď n { α , wherefore S is atmost p αn q {
6. Thus to conclude the argument it is more than enough to show S ď p αn q { W “ ř i P P α i we have ř i Pr n s (cid:114) P α i “ ´ W and S ď ÿ i P P α i ¨ ÿ jk P p r n s (cid:114) P q α j α k ď W ¨ p W { q “ W { , which by | W | ď α | P | ď αn completes the proof. (cid:3) YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 7
Proof of Proposition . Since the left side of ( ) is continuous in p x , . . . , x n q and ∆ n ´ is compact, there exists a point ξ “ p ξ , . . . , ξ n q P ∆ n ´ such that ω “ L G p ξ , . . . , ξ n q ` n ÿ i “ ˆ ξ i ´ n ˙ ´ ˆ ´ k ` n ` k ´ sn ˙ (2.3)is maximum. Assume for the sake of contradiction that ω ą Claim 2.4.
There exists an index i p‹q P r n s such that ξ i p‹q ą n ` sn .Proof of Claim . Define α , . . . , α n P r´ , n ´ s by ξ i “ p ` α i q{ n for every i P r n s andobserve that ωn “ L G p ` α , . . . , ` α n q ` n n ÿ i “ α i ´ | G |“ n ÿ i “ d G p i q α i ` ÿ ď i ă j ď n d G p i, j q α i α j ` L G p α , . . . , α n q ` n n ÿ i “ α i . Since all vertices of G have the same degree and ř ni “ α i “ n `ř ni “ ξ i ´ ˘ “
0, the first sumon the right side vanishes. Moreover, all pairs of vertices have codegree n ´ k in K n (cid:114) H p D q and thus we obtain ωn “ ˆ n ´ n ´ k ˙ n ÿ i “ α i ´ ÿ ď i ă j ď n d S p i, j q α i α j ` L G p α , . . . , α n q . (2.4) First case: We have ξ , . . . , ξ n ą . Collecting the quadratic and cubic terms in ( ) separately we put Q “ ˆ n ´ n ´ k ˙ n ÿ i “ α i ´ ÿ ď i ă j ď n d S p i, j q α i α j and K “ L G p α , . . . , α n q , so that ωn “ Q ` K. Now for every real number C sufficiently close to 1 the point p ξ , . . . , ξ n q defined by ξ i “ p ` Cα i q{ n belongs to ∆ n ´ and the maximal choice of ω reveals L G p ξ , . . . , ξ n q ` n ÿ i “ ˆ ξ i ´ n ˙ ´ ˆ ´ k ` n ` k ´ sn ˙ ď ω. Multiplying by n and repeating the above calculation we obtain QC ` KC ď Q ` K and thus 0 ď p ´ C qrp ` C q Q ` p ` C ` C q K s (2.5) XIZHI LIU, DHRUV MUBAYI, AND CHRISTIAN REIHER whenever | C ´ | is sufficiently small. Letting C tend to 1 from above and below we obtain2 Q ` K “
0. Substituting this back into ( ) we learn0 ď p ´ C qrp C ´ q Q ` p C ` C ´ q K s “ ´p ´ C q r Q ` p C ` q K s . Thus Q ` K ď p ´ C q K holds whenever | C ´ | is sufficiently small, which is onlypossible if Q ` K ď
0. Together with Q ` K “ ωn ą K ă ωn ă Q ´ K “ p´ q Q ` p´ q Q . So the maximality of ω tells us that for C “ ´ p ξ , . . . , ξ n q R ∆ n ´ . In other words, there is some i p‹q P r n s such that ξ i p‹q ě n ą n ` sn , as desired. Second case: There exists some j p‹q P r n s satisfying ξ j p‹q “ . Now α j p‹q “ ´ n ÿ i “ α i ě . (2.6)Next we observe that the hypothesis that S be s -regular yields ´ ÿ ď i ă j ď n d S p i, j q α i α j ď ÿ ď i ă j ď n d S p i, j q α i ` α j “ s n ÿ i “ α i . Combined with ( ) and the positivity of ω this shows ˆ n ´ k ´ n ´ s ˙ n ÿ i “ α i ă L G p α , . . . , α n q . (2.7)Due to n ě k ` s we have n ´ k ´ n ´ s ą ˆ ´ ´ ´ ˙ n “ n ą p s q and together with ( ), ( ) this establishes p s q ă L G p α , . . . , α n q . In view of Fact we deduce that α i p‹q ą s { n holds for some i p‹q P r n s and now ξ i p‹q “ ` α i p‹q n ą n ` sn follows. Thereby Claim is proved. (cid:3) Now for every i P r n s we set D i “ B L G p x , . . . , x n qB x i ˇˇˇˇ p ξ ,...,ξ n q “ ÿ jk P L i ξ j ξ k , YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 9 where L i denotes the link graph of i in G . Owing to the maximality of ω in ( ) theLagrange multiplier method leads to the existence of a real number M such that D i ` ˆ ξ i ´ n ˙ “ M holds for every vertex i P r n s with ξ i ą
0. Notice that M “ M n ÿ j “ ξ j “ n ÿ j “ ξ j ˆ D j ` ˆ ξ j ´ n ˙˙ “ L G p ξ , . . . , ξ n q ` n ÿ j “ ˆ ξ j ´ n ˙ ) ą ˆ ´ k ` n ` k ´ sn ˙ ´ n ÿ j “ ˆ ξ j ´ n ˙ . Altogether, this proves D i ` ˆ ξ i ´ n ˙ ` n ÿ j “ ˆ ξ j ´ n ˙ ą ˆ ´ k ` n ` k ´ sn ˙ for every vertex i P r n s satisfying ξ i ą K n (cid:114) H p D q of every vertex i P r n s is a q -partite Turán graph with vertex classes of size k ´
1, where q “ p n ´ q{p k ´ q is aninteger. Consequently, there exist real numbers β , . . . , β q such that ξ i ` p β ` ¨ ¨ ¨ ` β q q “ D i ď ÿ ď v ă w ď q β v β w ď q ´ q p β ` ¨ ¨ ¨ ` β q q “ n ´ k p n ´ q p ´ ξ i q . Summarizing, we have29 ˆ ξ i ´ n ˙ ` n ÿ j “ ˆ ξ j ´ n ˙ ą n ´ k p n ´ q ˜ˆ ´ n ˙ ´ p ´ ξ i q ¸ ´ sn (2.8)for every vertex of positive weight. For the rest of the argument we fix a vertex i p‹q P r n s such that ξ i p‹q is maximal. Let us add the trivial estimate19 n ÿ j “ ξ j p ξ i p‹q ´ ξ j q ě i “ i p‹q of ( ). Because of n ÿ j “ ˆ ξ j ´ n ˙ ` n ÿ j “ ξ j p ξ i p‹q ´ ξ j q “ n ÿ j “ ξ j ˆ ξ i p‹q ´ n ˙ ´ n n ÿ j “ ˆ ξ j ´ n ˙ (2.9) “ ξ i p‹q ´ n (2.10) this yields 13 ˆ ξ i p‹q ´ n ˙ ą n ´ k p n ´ q ˆ ξ i p‹q ´ n ˙ ˆ ´ n ´ ξ i p‹q ˙ ´ sn ě n ´ k n ˆ ξ i p‹q ´ n ˙ ´ sn ě ˆ ξ i p‹q ´ n ˙ ´ sn , whence ξ i p‹q ă n ` sn . Owing to the maximal choice of ξ i p‹q this contradicts Claim . (cid:3) § Constructions and Turán numbers
Given a positive integer t we define in this section the triple systems G , . . . , G t andthe forbidden family M t appearing in Theorem . For every i P r t s there will be threeintegers n i , k i , s i such that G i “ K n i (cid:114) p H p D i q Y S i q holds for some p n i , k i q -design D i on r n i s and some s i -regular triple system S i on r n i s that is disjoint to H p D i q . As we shallhave n i " k i , s i , Proposition will imply λ p G i q “ ˆ ´ k i ` n i ` k i ´ s i n i ˙ . Part of our goal is that balanced blow-ups of G , . . . , G t should be extremal M t -free triplesystems and for this reason we need to ensure λ p G i q “ ¨ ¨ ¨ “ λ p G t q . We shall achieve thisby letting k i “ s i for i P r t s , and by guaranteeing k ` n “ ¨ ¨ ¨ “ k t ` n t . (3.1)The details of this construction are given in Subsection and the exact Turán numbersof our families M t are determined in Subsection .3.1. The extremal configurations and forbidden family.
First, we need the followingtheorem about the existence of designs due to Wilson [ – ]. Theorem 3.1 (Wilson [ – ]) . For every integer k ě there exists a threshold n p k q such that for every integer n ě n p k q satisfying the divisibility conditions p k ´ q | p n ´ q and p k ´ q k | p n ´ q n there exists an p n, k q -design. Our next lemma deals with the arithmetic properties the numbers k , . . . , k t and n , . . . , n t entering the construction of G , . . . , G t need to satisfy. Apart from ( ) and the divisibilityconditions in Theorem we will require that n , . . . , n t are divisible by 3 so that p k i { q -regular triple systems on n i vertices exist. Thus the case q “ YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 11
Lemma 3.2.
Given positive integers t and q there exist t even integers ă k ă ¨ ¨ ¨ ă k t such that for every constant C ą there exist t integers n ă ¨ ¨ ¨ ă n t with the followingproperties. ( a ) We have q | n i , p k i ´ q | p n i ´ q , and k i p k i ´ q | n i p n i ´ q for all i P r t s . ( b ) Moreover, Q “ n k ` “ ¨ ¨ ¨ “ n t k t ` is an integer with Q ě C .Proof of Lemma . Starting with an arbitrary positive multiple s of q we recursivelydefine integers 1 ď s ă ¨ ¨ ¨ ă s t by setting s i ` “ ś j ď i s j p s j ´ q ` i P r t ´ s .Now whenever 1 ď i ă j ď t we have s j ” p mod s i p s i ´ qq and, consequently, s j p s j ´ q ” p mod s i p s i ´ qq . In particular, the numbers s p s ´ q , . . . , s t p s t ´ q are pairwise coprime and by the Chinese remainder theorem there exists an even integer Q ě C such that Q { ” s i p mod s i p s i ´ qq holds for all i P r t s . Multiplying thesecongruences by 2 and setting k i “ s i we obtain Q ” k i { p mod k i p k i ´ qq . (3.2)Now it is plain that the numbers n i “ Q p k i ` q satisfy ( b ) . Moreover, the case i “ )yields q | k | Q and, therefore, n , . . . , n t are divisible by q . Finally, multiplying ( ) by k i ` n i ” k i p k i ` qp k i { q ” k i p k i { q ” k i ” k i p mod k i p k i ´ qq , for which reason k i | n i and p k i ´ q | p n i ´ q . So altogether ( a ) holds as well. (cid:3) Given two r -graphs H and H with the same number of vertices a packing of H and H is a bijection ϕ : V p H q Ñ V p H q such that ϕ p E q R H for all E P H . In order toproceed with our construction of the triple systems G , . . . , G t we need to argue that, undernatural assumptions, if D i denotes an p n i , k i q -design, then there is an s i -regular 3-graph S i Ď K n (cid:114) H p D i q , where s i “ k i {
2. Provided that 3 | n i and s i ď ` n ´ ˘ the existence ofsome s i -regular 3-graph S i Ď K n is a well known fact that follows, e.g., from Baranyai’sfactorisation theorem [ ]. For making S i and H p D i q disjoint we use a packing argumentbased on the following result of Lu and Székely. Theorem 3.3 (Lu-Székely [ ]) . Let H and H be two r -graphs on n vertices. If ∆ p H q| H | ` ∆ p H q| H | ă er ˆ nr ˙ , then there is a packing of H and H . In fact, we only require the following consequence.
Corollary 3.4.
Suppose | n and that D is an p n, k q -design on r n s . If s ă n ´ e p k ´ q , thenthere exists an s -regular -graph S on r n s such that S X H p D q “ ∅ .Proof of Corollary . By 3 | n and s ď ` n ´ ˘ there is an s -regular 3-graph S on n vertices. Since∆ p S q| H p D q| ` ∆ p H p D qq| S | “ s k ´ n p n ´ q ` k ´ p n ´ q sn “ s k ´ n p n ´ q ă n ´ e p k ´ q k ´ n p n ´ q“ e ˆ n ˙ , Theorem yields a packing ϕ : V p S q Ñ r n s of S and H p D q . It is clear that S “ ϕ p S q satisfies the requirements of Corollary . (cid:3) Now we are ready to present the definition of G , . . . , G t . Construction 3.5.
Given a positive integer t perform the following steps. ‚ Apply Lemma with q “ , thus getting some even integers ă k ă ¨ ¨ ¨ ă k t . ‚ Take an integer C ě max t n p k q , . . . , n p k t q , k t , u , where the thresholds n p k i q are given by Theorem . ‚ Now Lemma applied to C and k , . . . , k t yields integers C ă n ă ¨ ¨ ¨ ă n k suchthat, in particular, Q “ n k ` “ ¨ ¨ ¨ “ n t k t ` is an integer with Q ě C .Now, for every i P r t s‚ let D i be an p n i , k i q -design on r n i s (as obtained by Theorem ) ‚ let S i be a p k i { q -regular -graph on r n i s such that S i X H p D i q “ ∅ (as obtained byCorollary ). ‚ and, finally, define G i “ K n i (cid:114) p H p D i q Y S i q . YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 13
By Proposition we have λ p G i q “ ˆ ´ k i ` n i ` k i ´ k i { n i ˙ “ ˆ ´ Q ˙ . for every i P r t s , so some rational λ t satisfies λ t “ λ p G q “ ¨ ¨ ¨ “ λ p G t q P r { , { q . (3.3)In the remainder of this subsection we introduce the family M t . For an r -graph H anda set S Ď V p H q we say that S is 2 -covered in H if for every pair of vertices in S there is anedge in H containing it. If this holds for S “ V p H q then H itself is said to be 2-covered.For all integers ‘ ą r ě K r‘ denote the family of r -graphs F with at most ` ‘ ˘ edges that contain a 2-covered set S of ‘ vertices called a core of F . The family K r‘ was first introduced by the second author [ ] in order to extend Turán’s theorem tohypergraphs. It also plays a key rôle in in the construction of the family M with twoextremal configurations in [ ]. In the present work, we also need the larger family p K r‘ defined to consist of all r -graphs F with at most ` ‘r ˘ edges that contain a 2-covered set S of ‘ vertices, which is again called a core of F .Let us recall that the transversal number of a hypergraph H is the nonnegative integer τ p H q “ min t| S | : S Ď V p H q and S X E ‰ ∅ for all E P H u . Note that if H is empty, then we can take S “ ∅ , whence τ p H q “ M t is defined as follows. Definition 3.6.
For every positive integer t the family M t consists of all -graphs F P Ť ‘ ď n t p K ‘ which do not occur as a subgraph in any blow-up of G , . . . , G t and which have acore S such that τ p F r S sq ě . We conclude this subsection with a simple sufficient condition for 3-graphs F P K n t ` guaranteeing that they are in M t (see Lemma below). For this purpose we require thefollowing observation analysing the extent to which τ p H q ě H . Fact 3.7. If r ě and H denotes an r -graph with τ p H q ě , then there is a subgraph H Ď H with at most r ` edges satisfying τ p H q ě .Proof. Pick two distinct edges E , E P H and write E X E “ t v , . . . , v m u , where0 ď m ď r ´
1. For every i P r m s the assumption that t v i u fails to cover H yields an edge E i P H such that v i R E i . Now H “ t E , E , E , . . . , E m u has the desired properties. (cid:3) Notice that the example H “ K rr ` shows that the bound | H | ď r ` Lemma 3.8.
Suppose that F is a -graph and that S Ď V p F q is a -covered set in F .If τ p F r S sq ě , then F contains a subgraph F such that F P K | S | and τ p F r S sq ě .Moreover, if ď s ď | S | , then F has a subgraph F P K s possessing a core S such that τ p F r S sq ě .Proof of Lemma . The case r “ yields a subgraph G of F r S s with at mostfour edges such that τ p G q ě
2. Notice that | G | ě |B G | ě
5. Since S is 2-covered in F ,we can choose for every pair uw P ` S ˘ (cid:114) B G an edge e uw P F containing u and w . Now F “ (cid:32) e uw : uw P ` S ˘ (cid:114) B G ( Y G has the properties that S is 2-covered in F and τ p F r S sq ě
2. Together with | F | ď ˆ ‘ ˙ ´ |B G | ` | G | ď ˆ ‘ ˙ ´ ` ă ˆ ‘ ˙ this proves F P K | S | . Moreover, if any s P r , | S |s is given, we can take a set S of size s with V p G q Ď S Ď S and apply the first part of the lemma to S rather than S . (cid:3) Lemma 3.9. If S denotes a core of F P K n t ` and τ p F r S sq ě , then F P M t .Proof. By the previous lemma and n t ě
12 there exists a set S Ď S such that | S | “ n t and τ p F r S sq ě
2. Since | F | ď ` n t ` ˘ ď ` n t ˘ , we can regard F as a member of p K n t withcore S and it remains to prove that F cannot be G i -colorable for any i P r t s . This is dueto the fact that the shadows of blow-ups of G i are complete n i -partite graphs, while S induces a K n t ` in B F . (cid:3) Turán numbers of M t . Having now introduced the main protagonists G , . . . , G t and M t we shall determine the extremal numbers ex p n, M t q in this subsection. Moreprecisely, setting M p n q “ max t| G | : G is G i -colorable for some i P r t s and v p G q “ n u for every positive integer n we shall prove the following result. Theorem 3.10.
The equality ex p n, M t q “ M p n q holds for all positive integers n . Notice that in view of Lemma and ( ) this implies ex p n, M t q ď λ t n for everypositive integer n . Moreover, whenever n is a multiple of n i for some i P r t s , the balancedblow-up of G i with factor n { n i exemplifies that this holds with equality. For these reasons,Theorem is stronger than Theorem ( a ) . Let us start with the lower boundon ex p n, M t q . Fact 3.11.
We have ex p n, M t q ě M p n q for every positive integer n . YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 15
Proof of . This is an immediate consequence of the fact that by Definition for every i P r t s all blow-ups of G i are M t -free. (cid:3) Our proof for the upper bound uses the Zykov symmetrization method [ ]. The applica-bility of this technique in the current situation hinges on the fact that if a hypergraph H is M t -free, then there is no homomorphism from a member of M t to H (see Proposition below). Let us recall that given two r -graphs F and H a map ϕ : V p F q ÝÑ V p H q is said tobe a homomorphism if ϕ preserves edges, i.e., if ϕ p E q P H holds for all E P F . Further, H is F -hom-free if there is no homomorphism from F to H or, in other words, if F fails to be H -colourable. For a family F of r -graphs, we say that H is F -hom-free if it is F -hom-freefor every F P F . Proposition 3.12. A -graph H is M t -hom-free if and only if it is M t -free.Proof of Proposition . Notice that the forward implication is clear. Now supposeconversely that H fails to be M t -hom-free, i.e., that there is a homomorphism ϕ : V p F q ÝÑ V p H q for some F P M t . Clearly the restriction of ϕ to a core S of F is injective. So ϕ p F q P p K | S | X M t and in view of ϕ p F q Ď H it follows that H fails to be M t -free. (cid:3) As an immediate consequence of Definition , semibipartite triple systems are M t -free.We analyze the semibipartite case as follows. Lemma 3.13. If H denotes a semibipartite triple system on n vertices, then | H | ď min t n { , M p n qu . Proof.
Fix a partition V p H q “ A Y¨ B such that | E X A | “ E P H . Nowthe AM-GM inequality yields | H | ď | A | ˆ | B | ˙ ď | A | ¨ | B | ¨ | B | ď ˆ | A | ` | B | ` | B | ˙ “ n | H | ď M p n q . If n is large this is an immediate consequence of M p n q “ p λ t ´ o p qq n and λ t ě { ą {
27, but for a complete proof addressing all valuesof n we need to argue more carefully.To this end we consider a random map ϕ : r n s ÝÑ r n s together with the randomblow-up p G of G determined by ϕ . Explicitly p G has vertex set r n s and a triple ijk forms anedge of p G if and only if ϕ p i q ϕ p j q ϕ p k q P G . Now every potential edge of p G is present withprobability | G i | n “ λ t and thus the expectation of | p G | is 6 λ t ` n ˘ . So by averaging we obtain M p n q ě λ t ˆ n ˙ ě ˆ n ˙ , (3.4) which for n ě M p n q ě n {
27. Moreover, ( ) yields M p q ě
3, which still suffices for the case n “ n ď (cid:3) The central notion in arguments based on Zykov symmetrization is the following:Given an r -graph H , two non-adjacent vertices u, v P V p H q are said to be equivalent if L H p u q “ L H p v q . Evidently, equivalence is an equivalence relation. Since any two equivalentvertices have the same degree and the same link, we can write d H p C q and L H p C q for thecommon degree and the common link of all vertices in an equivalence class C , respectively. Lemma 3.14.
Let H be an M t -free -graph with equivalence classes C , . . . , C m . If forall distinct k, ‘ P r m s the shadow B H induces a complete bipartite graph between C k and C ‘ ,then H is either semibipartite or G i -colourable for some i P r t s .Proof of Lemma . Let T Ď V p H q be a set containing exactly one vertex from eachequivalence class of H , and let T be the subgraph of H induced by T . By assumption, T is 2-covered, | T | “ m , and H is a blow-up of T . If τ p T q ă
2, then T is a star and H issemibipartite. So we may assume τ p T q ě T is 2-covered and | T | ď ` m ˘ we have T P p K m . So if m ď n t , then in view ofDefinition and T R M t there exists an index i P r t s such that T is G i -colorable. As H is a blow-up of T , it follows that H is G i -colorable as well.Now assume for the sake of contradiction that m ą n t . Since n t ě
12, Lemma leadsto a subgraph T P K n t ` of T having a core S such that τ p T r S sq ě
2. By Lemma this contradicts H being M t -free. (cid:3) Now we are ready to establish the main result of this subsection.
Proof of Theorem . Fix some positive integer n . By Fact it suffices to establishthe upper bound ex p n, M t q ď M p n q . Arguing indirectly we choose an M t -free triplesystem H on n vertices with more than M p n q edges such that the number m of equivalenceclasses of H is minimal. Let C , . . . , C m be the equivalence classes of H .By Lemma we know that H is not semibipartite and the definition of M p n q impliesthat H fails to be G i -colorable for every i P r t s . For these reasons, Lemma tells usthat B H is not the complete m -partite graph with vertex classes C , . . . , C m . Without lossof generality we may assume that at least one possible edge between C and C is missingin B H . Due to the definition of equivalence there are actually no edges between C and C in B H . By symmetry we may suppose further that d H p C q ď d H p C q .Now let H be the unique 3-graph satisfying V p H q “ V p H q , H ´ C “ H ´ C , and L H p v q “ L H p w q for all v P C and w P C . Observe that t C Y C , C , . . . , C m u refines the YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 17 partition of V p H q into the equivalence classes of H and | H | “ | H | ` | C | ` d H p C q ´ d H p C q ˘ ě | H | ą M p n q . So our minimal choice of m implies that H cannot be M t -free. As there exists a ho-momorphism from H to H , it follows that H fails to be M t -hom-free. But owing toProposition this contradicts H being M t -free. (cid:3) § Stability
In this section we prove most of Theorem ( b ) – only the proof of ξ p M t q “ t ispostponed to Section . Our goal is to show that after deleting a small number of low-degreevertices an “almost extremal” M t -free 3-graph becomes G i -colorable for some i P r t s . Moreprecisely, we aim for the following result. Theorem 4.1. If ε ą is sufficiently small, n is sufficiently large, and H is an M t -free -graph on n vertices with | H | ě p λ t ´ ε q n , then the set Z “ (cid:32) u P V p H q : d H p u q ď p λ t ´ ε { q n ( has size at most ε { n and the -graph H ´ Z is G i -colorable for some i P r t s . As the proof of this result will occupy the entire section, we would like to start with aquick overview. The argument is somewhat similar in spirit to [ , , ] and ultimately it isbased on the Zykov symmetrization method [ ]. There are certain kinds of complicationsthat often arise when one uses this strategy in order to establish stability results andwe overcome several of these common difficulties by introducing the so-called Ψ-trick inSubsection . By means of this trick, the problem to prove Theorem gets reduced toan apparently much simpler task: If a triple system H with n vertices and minimum degree p λ t ´ o p qq n can be made G i -colorable by deleting a single vertex, then, actually, H itselfis G i -colorable (see Lemma ). The Ψ-trick can also be used to reprove some knownstability results with improved control over the dependence of the constants (see [ ]).The proof of Lemma is still quite long. We will collect some auxiliary results inSubsection and defer the main part of the argument to Subsection General preliminaries.
This subsection reduces the task of proving Theorem to the apparently much simpler task of verifying Lemma below. There are only few“special properties” of M t we are going to utilize in the course of this reduction and werefer to [ ] for a more systematic treatment. Throughout this subsection we use the following notation: For every 3-graph H on n vertices and every ε ą Z ε p H q “ (cid:32) u P V p H q : d H p u q ď p λ t ´ ε { q n ( . Lemma 4.2. If ε P p , q , n ě ε ´ { and H is an M t -free -graph on n vertices with atleast p λ ´ ε q n edges, then ( a ) the set Z ε p H q has at most the size ε { n ( b ) and the subgraph H “ H ´ Z ε p H q of H satisfies δ p H q ě p λ t ´ ε { q n as well as | H | ě p λ t ´ ε { q n .Proof of Lemma . Set Z “ Z ε p H q . Assuming that part ( a ) fails we can take a set X Ď Z of size ε { n ď | X | ď ε { n . The definition of Z leads to | H ´ X | ě p λ t ´ ε q n ´ | X |p λ t ´ ε { q n ě p λ t ´ ε q n ´ | X |p λ t ´ ε { q n ´ n p| X | ´ ε { n qp ε { n ´ | X |q“ λ t p n ´ | X |q ` p { ´ λ t q n | X | ` λ t | X | ą λ t p n ´ | X |q , where we used λ t ă { ă { ( a ) thiscontradicts the fact that H ´ X is M t -free.Now we prove part ( b ) . For every u P V p H q the definition of Z and ( a ) yield d H p u q ě d H p u q ´ | Z | n ě p λ t ´ ε { q n ´ ε { n “ p λ t ´ ε { q n . Similarly, we have | H | ě | H | ´ | Z | n ě p λ t ´ ε q n ´ ε { n ą p λ t ´ ε { q n . (cid:3) The following lemma will be shown to imply Theorem . Lemma 4.3.
There exist constants ζ P p , q and N P N such that the following holds forall n ě N . Let H be an M t -free -graph on n vertices with at least p λ t ´ ζ q n edges and δ p H q ą p λ t ´ ζ q n . If there exists a vertex v P V p H q such that H ´ v is G i -colorable forsome i P r t s , then H itself is G i -colorable as well. We postpone the proof of this result to Subsection . The deduction of Theorem from Lemma factorises through the following statement.
Lemma 4.4.
There exists ε P p , { q such that the following holds for every sufficientlylarge integer n . Let H denote an M t -free -graph with n vertices and at least p λ t ´ ε q n edges. If Q Ď V p H q has size | Q | ď ε { n and H ´ Q is G i -colourable for some i P r t s , then H ´ Z ε p H q is G i -colourable as well. YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 19
Proof of Lemma using Lemma . We show that ε “ ζ {
25 has the desired property,where ζ denotes the constant provided by Lemma . Given a sufficiently large 3-graph H and a set Q as described in the statement of Lemma we set Q “ Q (cid:114) Z ε p H q and V “ V p H q (cid:114) p Z ε p H q Y Q q .By our assumption, there is an index i p‹q P r t s such that H r V s is G i p‹q -colorable. Choosea set S Ď Q of maximum size such that H r V Y S s is still G i p‹q -colorable. If S “ Q we aredone, so suppose for the sake of contradiction that there exists a vertex v P Q (cid:114) S .Due to the maximality of S the triple system H “ H r V Y S Y t v us is not G i p‹q -colorable.On the other hand, Lemma ( a ) and | Q | ď ε { n entail δ p H q ą p λ t ´ ε { q n ´ | Z p H q Y Q | n ą p λ t ´ ε { q n and | H | ą p λ t ´ ε q n ´ | Z p H q Y Q | n ą p λ t ´ ε { q n . So by Lemma and ζ “ ε { the G i p‹q -colorability of H ´ v “ H r V Y S s implies that H itself is G i p‹q -colorable as well. This contradiction completes the proof of Lemma (cid:3) It remains to deduce Theorem . The argument involves the following invariant of3-graphs: Given a 3-graph H with equivalence classes C , . . . , C m we set Ψ p H q “ ř mi “ | C i | . Proof of Theorem using Lemma . Let ε be the constant delivered by Lemma and fix a sufficiently large natural number n . Assuming that the conclusion of Theorem fails for our values of ε and n we pick a counterexample H such that the pair p| H | , Ψ p H qq is lexicographically maximal. Let C , . . . , C m be the equivalence classes of H .Recall that Lemma ( a ) tells us | Z ε p H q| ď ε { n . Since H is a counterexample, itcannot be G i -colorable for any i P r t s . Moreover, ( ) yields | H | ą p λ t ´ ε q n ě p { ´ { q n “ n { ą n { H cannot be semibipartite. So by Lemma there exist two equivalence classes,say C and C , such that B H possesses no edges from C to C . We may assume that p d H p C q , | C |q ď lex p d H p C q , | C |q , where ď lex indicates the lexicographic ordering on N .Pick arbitrary vertices v P C and v P C and symmetrize only them. That is, we let H be the 3-graph with V p H q “ V p H q , H ´ v “ H ´ v and L H p v q “ L H p v q . Clearly,if d H p v q ă d H p v q , then | H | ą | H | . Moreover, if d H p v q “ d H p v q , then | H | “ | H | , | C | ď | C | , andΨ p H q ´ Ψ p H q ě p| C | ´ q ` p| C | ` q ´ | C | ´ | C | “ p| C | ´ | C | ` q ě . In both cases p| H | , Ψ p H qq is lexicographically larger than p| H | , Ψ p H qq and our choiceof H implies that H ´ Z ε p H q is G i -colourable for some i P r t s . By Lemma ( a ) the set Q “ Z ε p H q Y t v u has size | Q | ď ε { n ` ă ε { n . Since the hypergraph H ´ Q “ H ´ Q is G i -colourable, Lemma implies that H ´ Z p H q is G i -colourable too. This contradictionto the choice of H establishes Theorem . (cid:3) Transversals.
Roughly speaking, the hypergraph H ´ v appearing in Lemma arises from an almost balanced blow-up of G i by deleting a small number of edges. Whenwe randomly select one vertex from each partition class of H ´ v it is thus very likely thatthe resulting transversal induces a copy of G i . In the proof of Lemma there are severalplaces where we argue similarly in situations where some vertices from the transversals havebeen selected in advance. The precise statement we shall use in these cases is Lemma below.Consider a 3-graph with V p G q “ r m s and pairwise disjoint sets V , . . . , V m . The blow-up G r V , . . . , V m s of G is obtained from G by replacing each vertex j P r m s with the set V j andeach edge t j , j , j u P G with the complete 3-partite 3-graph with vertex classes V j , V j ,and V j . For a 3-graph H we say that a partition V p H q “ Ť ¨ j Pr m s V j is a G -coloring of H if H Ď G r V , . . . , V m s . Lemma 4.5.
Fix a real η P p , q and integers m, n ě . Let G be a -graph with vertexset r m s and let H be a further -graph with v p H q “ n . Consider a vertex partition V p H q “ Ť ¨ i Pr m s V i and the associated blow-up p G “ G r V , . . . , V m s of G . If two sets T Ď r m s and S Ď Ť j R T V j have the properties ( a ) | V j | ě p| S | ` q| T | η { n for all j P T , ( b ) | H r V j , V j , V j s| ě | p G r V j , V j , V j s| ´ ηn for all t j , j , j u P ` T ˘ , ( c ) and | L H p v qr V j , V j s| ě | L p G p v qr V j , V j s| ´ ηn for all v P S and t j , j u P ` T ˘ ,then there exists a selection of vertices u j P V j for all j P r T s such that U “ t u j : j P T u satisfies p G r U s Ď H r U s and L p G p v qr U s Ď L H p v qr U s for all v P S . In particular, if H Ď p G ,then p G r U s “ H r U s and L p G p v qr U s “ L H p v qr U s for all v P S .Proof of Lemma . Choose for j P T the vertices u j P V j independently and uniformlyat random and let U “ t u j : j P T u be the random transversal consisting of these vertices.By ( a ) and ( b ) we have P pt u j , u j , u j u R H q “ ´ | H r V j , V j , V j s|| V j || V j || V j | ď ηn | V j || V j || V j | ď p| S | ` q | T | for all edges t j , j , j u P G . Similarly ( a ) and ( c ) lead to P ` t u j , u j u R L H p v q | t u j , u j u P L p G p v q ˘ “ ´ | L H p v qr V j , V j s|| V j || V j | ď ηn | V j || V j |ď η { p| S | ` q | T | YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 21 for all v P S and all distinct j , j P r m s . Therefore, the union bound reveals P ´ p G r U s Ę H r U s ¯ ď ˆ | T | ˙ p| S | ` q | T | ă P ` L p G p v q Ę L H p v q ˘ ď ˆ | T | ˙ η { p| S | ` q | T | ă p| S | ` q for every v P S. Altogether, the probability that U fails to have the desired properties is at most16 ` | S | p| S | ` q ă . So the probability that U has these properties is positive. (cid:3) In practice the sets U obtained by means of Lemma will be 2-covered and thus theywill be cores of some subgraphs F P p K | U | of H . In such situations F will be M t -free andin order to exploit this fact we need to know that for i ‰ j the triple system G i is in somesense far from being G j -colorable (see Lemma below). The verification of this statementrequires that we take a closer look into Construction and the observation that followssummarizes everything we need in the sequel. Observation 4.6.
The triple systems G , . . . , G t have the following properties. ( a ) For i P r t s and v P G i the clique number ω p L G i p v qq of the link graph L G i p v q satisfies n i ´ k i ´ ´ k i ď ω p L G i p v qq ď n i ´ k i ´ . ( b ) We have n i ´ k i ´ ´ n i ` ´ k i ` ´ ą Qk i for every i P r t ´ s , where Q “ n k ` “ ¨ ¨ ¨ “ n t k t ` ě k t ě . ( c ) For i P r t s the -graph G i is regular with degree λ t n i and n i { ď n i ´ k i { ď δ p G i q ď ∆ p G i q ď n i ´ k i . Proof.
Part ( a ) follows from the fact that due to G i “ p K n i (cid:114) H p D i qq (cid:114) S i the link L G i p v q arises from an pp n i ´ q{p k i ´ qq -partite Turán graph by the deletion of k i { proof of part ( c ) is similar. For part ( b ) it suffices to calculate n i ´ k i ´ ´ n i ` ´ k i ` ´ “ Q p k i ` q ´ k i ´ ´ Q p k i ` ` q ´ k i ` ´ “ p Q ´ q ˆ k i ´ ´ k i ` ´ ˙ ě Q ˆ k i ´ ´ k i ˙ ą Qk i . (cid:3) As indicated earlier, this has the following consequence.
Lemma 4.7. If i P r t s and the triple system G i arises from G i by the deletion of at most Q {p k i q vertices, then G i fails to be G j -colorable for every j P r t s (cid:114) t i u .Proof of Lemma . Suppose first that j P r i ´ s . Due to δ p G i q ě δ p G i q ´ Q k i ě G i is 2-covered. Together with v p G i q “ n i ´ Q {p k i q ą Q p k i ` q ´ Q ě Q p k j ` q “ n j it follows that G i is indeed not G j -colorable.If j P p i, t s we take an arbitrary vertex v P V p G i q . The parts ( a ) and ( b ) of Observation yield ω ` L G i p v q ˘ ě ω p L G i p v qq ´ Q k i ě n i ´ k i ´ ´ k i ´ Q k i ą n j ´ k j ´ . On the other hand, by Observation ( a ) again, any G j -coloring of G i would show that ω ` L G i p v q ˘ ď ω ` L G j p v q ˘ ď n j ´ k j ´ . (cid:3) On most occasions the following corollary of Lemma will suffice.
Corollary 4.8. If i P r t s , the -graph H is M t -free and U Ď V p H q denotes a -coveredset of size n i ` , then H r U s is G i -free.Proof. Assume for the sake of contradiction that H r U s has a subgraph isomorphic to G i .If i ă t we can take a subgraph F P p K n i ` of H with F r U s “ H r U s having U as a core.As H r U s contains a copy of G i , we have τ p F r U sq ě
2. Now F R M t implies that F is G j -colorable for some j P r t s . In particular, G i is G j -colorable and by Lemma this leadsto i “ j . In other words, F is G i -colorable, contrary to the fact that B F contains a copyof K n i ` . YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 23
It remains to discuss the case i “ t . Now Lemma yields a subgraph F of F whichbelongs to K n t ` , and whose induced subgraph on its core has covering number at least 2.By Lemma this contradicts H being M t -free. (cid:3) Proof of the main lemma.
This entire subsection is devoted to the proof ofLemma . Select constants ζ and N fitting into the hierarchy N ´ ! ζ ! n ´ t . Consider an M t -free 3-graph H on n ě N vertices satisfying | H | ě p λ t ´ ζ q n and δ p H q ě p λ ´ ζ q n such that for some v P V p H q and i P r t s the 3-graph H v “ H (cid:114) t v u is G i -colorable. Set V “ V p H q and fix a partition Ť ¨ i Pr n i s V i “ V (cid:114) t v u exemplifying the G i -colorability of H v . We divide the argument that follows into three main parts each ofwhich consists of several claims. Part I. Analysis of H v . The three claims that follow only deal with H v but say nothingabout v and its link. Claim 4.9.
We have | V j | “ n { n i ˘ ζ { n for every j P r n i s .Proof of Claim . Set x j “ | V j |{p n ´ q for every j P r n i s . By Proposition (and theproof of Lemma ) we obtain | H v | “ L G i p x , . . . , x n i qp n ´ q ď ˜ λ t ´ ÿ j Pr n i s ˆ x j ´ n i ˙ ¸ n . Combined with | H v | ě p λ t ´ ζ q n ´ d H p v q ą p λ t ´ ζ q n this leads to ř j Pr n i s p x j ´ { n i q ď ζ , whence x j “ { n i ˘ p ζ q { and ˇˇ | V j | ´ n { n i ˇˇ ď p n ´ q ˇˇ x j ´ { n i ˇˇ ` { n i ď p ζ q { n ` { n i ď ζ { n. (cid:3) Recall that the sets V , . . . , V n i have been chosen in such a way that H v is a subgraphof the blow-up p G i “ G i r V , . . . , V n i s of G i . Our next objective is to compare the links ofan arbitrary vertex u P V (cid:114) t v u in H v and in p G . As a consequence of H v Ď p G i we know L H v p u q Ď L p G i p u q and | L H v p u q| ď | L p G i p u q| . Members of L p G i p u q (cid:114) L H v p u q are referred to asthe missing pairs of u . By Lemma the global number of missing edges can be boundedfrom above by ˇˇ p G i (cid:114) H v ˇˇ ď λ t p n ´ q ´ p λ t ´ ζ q n ` d H p v q ď ζn . (4.1)Locally we obtain the following. Claim 4.10.
Every u P V (cid:114) t v u satisfies | L p G i p u q| ă p λ t ` n i ζ { q n . Moreover the numberof missing pairs of u is bounded by | L p G i p u q (cid:114) L H v p u q| ă ζ { n i n .Proof of Claim . Since G i is p λ t n i q -regular, Claim yields ˇˇ L p G i p u q ˇˇ ď λ t n i ˆ nn i ` ζ { n ˙ “ λ t n ` ` ζ { n i ˘ ă p λ t ` ζ { n i q n , where we used λ t ă { ζ ! n ´ i . Owing to the minimum degreecondition δ p H q ě p λ t ´ ζ q n this entails the upper bound ˇˇ L p G i p u q (cid:114) L H v p u q ˇˇ ď ` λ t n ` ζ { n i n ˘ ´ ` λ t n ´ ζn ´ n ˘ ă ζ { n i n on the number of missing pairs of u . (cid:3) It can now be shown that in H v all neighborhoods have roughly the expected size n i ´ n i n ,but for our concerns it suffices to establish a lower bound. Claim 4.11.
We have | N H v p u q| ě n i ´ n i n ´ ζ { n i n for every u P V (cid:114) t v u .Proof of Claim . Let j P r n i s be the index satisfying u P V j . Since every vertex in V (cid:114) p V j Y N H v p u q Y t v uq belongs to at least δ p G q ¨ min t| V ‘ | : ‘ P r n i su missing pairs of u ,and every missing pair is counted at most twice in this manner, Claim yields ˇˇ V (cid:114) p V j Y N H v p u q Y t v uq ˇˇ ¨ δ p G q ¨ min (cid:32) | V ‘ | : ‘ P r n i s ( ă ζ { n i n . So by Observation ( c ) and Claim the assumption | N H v p u q| ă n i ´ n i n ´ ζ { n i n would yield the contradiction ` ζ { n i n ´ ζ { n ´ ˘ ¨ n i ¨ ˆ nn i ´ ζ { n ˙ ă ζ { n i n . Thereby Claim is proved. (cid:3)
Part II. Choice of a vertex class for v . Our strategy for showing that H is G i -colorableis to adjoin v to one the partition classes V , . . . , V n t . In fact, there is only one of theseclasses v fits into. Before finding this class we show a statement that has to hold if ourplan is sound. Claim 4.12.
We have L H p v q X ` V j ˘ “ ∅ for every j P r n i s .Proof of Claim . Without loss of generality we may assume that j “
1. Let u , u P V be two distinct vertices. By Lemma applied to S “ t u , u u and T “ r , n i s there existvertices u j P V j for j P r , n i s such that the subgraphs of H induced by t u , u , . . . , u n i u and t u , u , . . . , u n i u are isomorphic to G i . Now Corollary informs us that the set U “ t u , u , . . . , u n t u cannot be 2-covered, for which reason u u R B H . So, in particular,we have u u R L H p v q . (cid:3) YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 25
Claim 4.13.
There exists j P r n i s such that | N H p v q X V j | ă ζ { n .Proof of Claim . Suppose for the sake of contradiction that the sets W j “ N H p v q X V j satisfy | W j | ě ζ { n for every j P r n i s . Applying Lemma to W j here in place of V j there and to S “ ∅ , T “ r n i s we obtain vertices u j P V j for all j P r n i s such that the set U “ t u , . . . , u n i u induces a copy of G i in H . But now the 2-covered set U Y t v u contradictsCorollary . (cid:3) It will turn out later that the index j delivered by Claim is unique. Without loss ofgenerality we may assume that | N H p v q X V | ă ζ { n. (4.2) Part III. The link of v . It remains to show that L H p v q Ď L p G i p V q . To this end we define N v p u q “ (cid:32) j P r n i s : | N H p u, v q X V j | ě ζ { n ( for every u P N H p v q . The upper bound on ∆ p G i q in Observation ( c ) transfers to thesesets as follows. Claim 4.14.
We have | N v p u q| ď n i ´ k i for every u P N H p v q .Proof of Claim . Assume for the sake of contradiction that there is a set N ‹ Ď N v p u q such that | N ‹ | “ n i ´ k i ` ă n i ´
2. As in the proof of Claim there exist ver-tices u j P N H p u, v q X V j for j P N ‹ such that G i r N ‹ s is isomorphic to H r U s , where U “ t u j : j P N ‹ u .Now we consider the 3-graph F “ H r U Y t u, v us . Clearly U Y t u, v u is 2-covered in F and τ p F q ě τ p G i r N ‹ sq ě
2. So F R M t tells us that F is G s -colorable for some s P r t s .On the other hand by Lemma and | U | ě n i ´ k i ` ą n i ´ Q {p k i q the subgraph F r U s of F cannot be G s -colorable for any s P r t s (cid:114) t i u .Summarizing this discussion, F is G i -colorable. As F is also 2-covered, F is actuallyisomorphic to a subgraph of G i and, consequently, n i ´ k i ă | N ‹ | “ d F p u, v q ď ∆ p G i q ,contrary to Observation ( c ) . (cid:3) Claim 4.15.
We have | N H p v q X V j | ě ζ { n for every j P r , n i s .Proof of Claim . The minimum degree condition imposed on H and 6 λ t “ ´ k i ` n i yield ˆ ´ k i ` n i ´ ζ ˙ n “ p λ t ´ ζ q n ď d H p v q ď ∆ ` L H p v q ˘ | N H p v q| . Claim allows us to bound the first factor on the right side from above by∆ ` L H p v q ˘ ď p n i ´ k i q ˆ nn i ` ζ { n ˙ ` k i ζ { n ă n i ´ k i n i n ` k i ζ { n. Altogether we obtain n i ´ p k i ` q ´ n i ζ p n i ´ k i q ` k i n i ζ { ď | N H p v q| n , which due to n i ´ p k i ` q n i ´ k i “ ´ n i ´ k i ą ´ { n i and ζ ! n ´ i implies ˆ ´ { n i ˙ n ď | N H p v q| . On the other hand, setting I “ (cid:32) j P r , n i s : | N H p v q X V j | ě ζ { n ( Claim and ( ) leadto | N H p v q| ď | I | ˆ n i ` ζ { ˙ n ` ζ { n i n. Combining both estimates we arrive at | I | ą n i ´ {
4, whence I “ r , n i s . (cid:3) Claim 4.16.
We have N H p v q X V “ ∅ .Proof of Claim . Suppose that there exists u P N H p v q X V . Owing Claim we canapply Lemma with S “ t u u and T “ r , n i s in order to obtain vertices u j P N H p v q X V j for j P r , n i s such that H induces a copy of G i on U “ t u , . . . , u n i u . Since U Y t v u is2-covered, this contradicts Corollary . (cid:3) Let us recall that L p G i p V q denotes the common p G i -link of all vertices in V . Claim 4.17.
We have L H p v q Ď L p G i p V q .Proof of Claim . Due to the Claims and we know that L H p v q is an p n i ´ q -partite graph with vertex classes V , . . . , V n i . So if Claim fails we may assume withoutloss of generality 123 R G i and that there exists a pair u u P L H p v q with u P V , u P V .Since | V | ą n {p n i q and | N H p v q X V j | ě ζ { n for j P r , n i s , Lemma applied to S “ t u , u u and T “ t , , . . . , n i u delivers vertices u P V and u j P N H p v q X V j for j P r , n i s such that the set U “ t u , u , . . . , u n i u satisfies H r U s “ p G i r U s and L H p u ‘ qr U s “ L p G i p u ‘ qr U s for ‘ “ , . (4.3)Consider the set U “ t u , . . . , u n i u . Because of ( ) and 123 R G i the map i ÞÝÑ u i is anembedding of L G i p q into H and for this reason we have d H r U s p u q ě d G i p q . (4.4) YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 27
Next we choose for every j P r , n i s an edge e j P H such that u j , v P e j and observethat U is 2-covered in the 3-graph F “ t vu u u Y t e j : 4 ď j ď n i u Y H r U s . Moreover, | F | ď | G i | ` n i ´ ă ` n i ˘ implies F P p K n i . Since F r U s “ H r U s is isomorphicto G i ´ t , u , Lemma tells us that F cannot be G j -colorable for any j P r t s (cid:114) t i u . Buton the other hand we have τ p F r U sq ě F R M t , so altogether F is G i -colorable.Fix a homomorphism ϕ : V p F q ÝÑ V p G i q from F to G i . Since U and U v “ U Y t v u (cid:114) t u u are 2-covered subsets of F whose size is n i “ v p G i q , the map ϕ has to be bijective on U and U v , which is only possible if ϕ p v q “ ϕ p u q . Now ϕ embeds the link L F r U s p u q intothe link L G i p ϕ p u qq . Moreover, vu u P F implies that ϕ p u q ϕ p u q belongs to the link L G i p ϕ p u qq as well and by 123 R G i this edge is not in the image ϕ p L F r U s p u qq . Altogetherthis proves d F r U s p u q ` ď d G i p ϕ p u qq , which in view of F r U s “ H r U s and ( ) contradictsthe regularity of G i . (cid:3) By Claim the partition Ť j Pr n i s p V j , where p V j “ $&% V Y t v u if j “ V j if 2 ď j ď n i is a G i -coloring of H . This completes the proof of Lemma . § Feasible region of M t and ξ p M t q We prove Theorem and that ξ p M t q “ t in this section. First, let us show a simplelemma. Lemma 5.1.
Suppose that H is an n -vertex G i -colorable -graph for some i P r t s . If | H | ě p λ t ´ ε q n , then |B H | ě ` n i ´ n i ´ ε { n i ˘ n .Proof of Lemma . Let V p H q “ Ť j Pr n i s V j be a G i -coloring of H . Now by Proposition , | V j | “ p { n i ˘ ε { q n for all j P r n i s . Call a pair t u, v u with u P V j , v P V k and j ‰ k missing if uv R B H , and let M denote the set of all missing pairs. Since δ p G i q ě n i {
8, weobtain | M | ¨ n i ¨ ˆ n i ´ ε { ˙ n ď εn , which yields | M | ă εn . Therefore, |B H | ą ˆ n i ˙ ˆ ˆ n i ´ ε { ˙ n ´ | M | ą n i ´ n i n ´ ε { n i n . (cid:3) We remark that the stronger conclusion |B H | ě ` n i ´ n i ´ εn i ˘ n could be shown byarguing more carefully, but this is immaterial to what follows. Proof of Theorem . Recall from Section that semibipartite 3-graphs are M t -free.This yields projΩ p M t q “ r , s , as for every x P r , s there exists a good sequence ofsemibipartite 3-graphs such that the edge densities of their shadows converges to x .Theorem ( a ) implies that g p M t , x q ď λ t for all x P r , s . Furthermore for every i P r t s the sequence of balanced blow-ups of G i shows the equality g p M t , ´ { n i q “ λ t . So,in order to finish the proof it suffices to show that if some x P r , s satisfies g p M t , x q “ λ t ,then there is an index i P r t s such that x “ ´ { n i .Fix such an x P r , s and let p H n q n “ be a good sequence of M t -free 3-graphs realizing p x, λ t q . Consider an arbitrary δ ą ε ą , N be the constants guaranteedby Theorem ( b ) . Without loss of generality we may assume ε ď δ . By our choiceof p H n q n “ there exists n P N such that d p H n q “ λ t ˘ ε and d pB H n q “ x ˘ ε hold for all n ě n . By Theorem ( b ) , for every n ě max t n , N u the 3-graph H n is G i -colorable for some i “ i p n q P r t s after removing at most δv p H n q vertices. Therefore, |B H n | ď ˆ n i ´ n i ` δ ˙ v p H n q , and, on the other hand, by Lemma , |B H n | ą ˆ n i ´ n i ´ ε { n i ˙ p ´ δ q v p H n q ą n i ´ n i v p H n q ´ ` ε { n i ` δ ˘ v p H n q . Summarizing and taking ε ď δ into account we arrive at n i ´ n i ´ ` δ { n t ` δ ˘ ă d pB H n q ď n i ´ n i ` δ, (5.1)where, let us recall, i “ i p n q might depend on n . So what ( ) means is that if we set I i p δ q “ „ n i ´ n i ´ δ { n i ´ δ, n i ´ n i ` δ for every i P r t s , then d pB H n q P I p δ q Y ¨ ¨ ¨ Y I t p δ q holds for every n ě n . As the set on the right side is closed we obtain x P I p δ q Y ¨ ¨ ¨ Y I t p δ q in the limit n Ñ 8 . Since δ ą x P č δ ą ` I p δ q Y ¨ ¨ ¨ Y I t p δ q ˘ “ (cid:32) ´ { n i : i P r t s ( YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 29 follows. (cid:3)
Recall that we already proved that M t is t -stable, which, by definition, shows that ξ p M t q ď t . Therefore, in order to prove ξ p M t q “ t it suffices to show that ξ p M t q ě t , andthis is an easy consequence of the following proposition and Theorem . Proposition 5.2.
Let F be a family of r -graphs and let M be the set of global maximaof g p F q . If M is finite, then | M | ď ξ p F q . The proof of this result involves the edit distance of hypergraphs: Given two r -graphs H and H with the same number of vertices we set d p H, H q “ min t| H H | : V p H q “ V p H q and H – H u . It is well known and easy to confirm that this distance satisfies the triangle inequality.
Proof of Proposition . If F is degenerate, then g p F q is the constant function whosevalue is always 0 and M is infinite. So we may assume that the Turán density y “ π p F q is positive. Let us write M “ tp x i , y q : i P r m su such that x ă ¨ ¨ ¨ ă x m and m “ | M | .For every i P r m s we select a good sequence p H i p n qq n “ of F -free r -graphs realizing p x i , y q .Without loss of generality we have v p H i p n qq “ n for every positive integer n . Now supposefor the sake of contradiction that t “ ξ p F q is smaller than m . Claim 5.3.
For every δ ą there are distinct i, j P r m s and n ą { δ such that d p H i p n q , H j p n qq ď δn r and min t| H i p n q| , | H j p n q|u ě p y ´ δ q ˆ nr ˙ . Proof of Claim . By the definition of ξ p F q “ t there are n P N and ε ą n ě n there exists a family t G p n q , . . . , G t p n qu of r -graphs on n vertices such thatfor every F -free r -graph H with v p H q “ n and | H | ě p y ´ ε q ` nr ˘ there is some s P r t s suchthat d p H , G s p n qq ď p δ { q n r As usual, we may suppose that ε ď δ .Now choose n ě n , δ ´ such that for every i P r m s we have d p H i p n qq ě y ´ ε . Stabilityallows us to select for every i P r m s an index s p i q P r t s such that d p H i p n q , G s p n qq ď δn r .By t ă m the map i ÞÝÑ s p i q cannot be injective, i.e., there are distinct i, j P r m s and s P r t s such that s p i q “ s p j q “ s . Now the triangle inequality yields d p H i p n q , H j p n qq ď d p H i p n q , G s p n qq ` d p G s p n q , H j p n qq ď δn r , as desired. (cid:3) Notice that, as stated, Claim allows i and j to depend on δ . However, a quickthought reveals that there actually have to be two indices i ă j that work for every δ ą M by proving r x i , x j s ˆ t y u Ď M . To this end, let x P r x i , x j s and a large integer N be given. It suffices to constructan F -free r -graph H satisfying v p H q ą N , d pB H q “ x ˘ { N and d p H q “ y ˘ { N . ByClaim applied to δ ! N ´ there is some n ą N such that d p H i p n q , H j p n qq ď δn r andmin t| H i p n q| , | H j p n q|u ě p y ´ δ q ` nr ˘ . Assume without loss of generality that | H i p n q j p n q| ď δn r . Now consider the following process transforming H i p n q into H i p n q : Start with H i p n q and remove edges one by one until H i p n q X H j p n q is reached. Then, keep adding edgesone by one until you arrive at H j p n q . Every r -graph occurring along the way is F -free.Moreover, since deleting or adding an edge can affect the size of the shadow by at most r ,in every step of the process the shadow density changer by at most r { ` nr ´ ˘ . Thus atsome moment we pass an r -graph H such that | d pB H q ´ x | ď r { ` nr ´ ˘ ď δ . Finally, d p H q ě d p H i p n q X H j p n qq ě d p H i p n qq ´ | H i p n q j p n q|{ ` nr ˘ ě y ´ O p δ q completes theproof that H has all desired properties. (cid:3) § Concluding remarks
For every positive integer t we constructed a family of 3-graphs t G , . . . , G t u that havethe same Lagrangian λ t , and we showed that there is a family M t of 3-graphs whoseextremal configurations are balanced blow-ups of G , . . . , G t , and whose stability numberis ξ p M t q “ t . Notice that our choice of λ t is very close to 1 {
6, which is the supremumof the Lagrangians of all 3-graphs. It would be interesting to find for every integer t ě λ “ λ p t q so that there exists a t -stable family F t with π p F t q “ λ . A result of Erdős [ ] implies that there are no Turán densities in the interval p , { q . This motivates the following question. Problem 6.1.
Does there exist a family F of triple systems with π p F q “ { but ξ p F q ‰ ? For a family F of r -graphs let M p F q “ t x P projΩ p F q : g p F qp x q “ π p F qq be the set ofabscissae of the global maxima of its feasible region function. As we have shown here, | M p F q| can be every finite cardinal except zero. In would be interesting to know whether M p F q can be infinite and, in case the answer is affirmative, there immediately arise furtherquestions. Problem 6.2.
For r ě does there exist a non-degenerate family F of r -graphs so that g p F q has infinitely many global maxima? If so, can the set M p F q be uncountable? Can iteven contain a non-trivial interval? YPERGRAPHS WITH MANY EXTREMAL CONFIGURATIONS 31
Notice that if the last question on intervals has a negative answer, then in Proposition the assumption that M should be finite can be omitted. In fact, it is somewhat bizarrethat we do not know the following. Problem 6.3.
Let F be a non-degenerate family of r -graphs such that M p F q is infinite.Can it nevertheless happen that F has finite stability number? In a forthcoming work [ ] we will show an extension of our results about triplessystems to r -graphs for all r ě M rt that is t -stable such that thefunction g p M rt q has exactly t -global maxima. References [1] Zs. Baranyai,
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Email address : [email protected] Department of Mathematics, Statistics, and Computer Science, University of Illinois,Chicago, IL 60607 USA
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