Erdos-Hajnal for graphs with no 5-hole
aa r X i v : . [ m a t h . C O ] F e b Erd˝os-Hajnal for graphs with no 5-hole
Maria Chudnovsky Princeton University, Princeton, NJ 08544Alex Scott Mathematical Institute, University of Oxford, Oxford OX2 6GG, UKPaul Seymour Princeton University, Princeton, NJ 08544Sophie Spirkl University of Waterloo, Waterloo, Ontario N2L3G1, Canada Supported by NSF grant DMS 1763817. Research supported by EPSRC grant EP/V007327/1. Supported by AFOSR grant A9550-19-1-0187, and by NSF grant DMS-1800053. We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada(NSERC), [funding reference number RGPIN-2020-03912]. Cette recherche a ´et´e financ´ee par le Conseil derecherches en sciences naturelles et en g´enie du Canada (CRSNG), [num´ero de r´ef´erence RGPIN-2020-03912]. bstract
The Erd˝os-Hajnal conjecture says that for every graph H there exists τ > G not containing H as an induced subgraph has a clique or stable set of cardinality at least | G | τ .We prove that this is true when H is a cycle of length five.We also prove several further results: for instance, that if C is a cycle and H is the complementof a forest, there exists τ > G containing neither of C, H as an inducedsubgraph has a clique or stable set of cardinality at least | G | τ . Introduction
A cornerstone of Ramsey theory is the theorem of Erd˝os and Szekeres [14] from the 1930s, that everygraph on n vertices has a clique or stable set of size Ω(log n ). This order of magnitude cannot beimproved, as Erd˝os [11] showed that there are infinitely many graphs G with max( α ( G ) , ω ( G )) = O (log( | G | )), where α ( G ) and ω ( G ) denote the cardinalities of (respectively) the largest stable setsand cliques in G . Indeed, for most graphs G , both α ( G ) and ω ( G ) are of logarithmic size.The celebrated Erd˝os-Hajnal conjecture asserts that for proper hereditary classes of graphs, thepicture is dramatically different. We say that a graph G contains a graph H if some induced subgraphof G is isomorphic to H , and G is H -free otherwise. Every hereditary class of graphs is defined byits excluded subgraphs (that is, the graphs F such that every graph in the class is F -free). TheErd˝os-Hajnal conjecture [12, 13] asserts that if some graph is excluded, the largest clique or stableset that can be guaranteed jumps from logarithmic to polynomial size: For every graph H , there exists τ > such that every H -free graph G satisfies max( α ( G ) , ω ( G )) ≥ | G | τ . The Erd˝os-Hajnal conjecture is only known for a small family of graphs. It is trivially true for H = K ; it is true for H = P , the four-vertex path (the P -free graphs form the well-known classof cographs); and Chudnovsky and Safra [7] showed that it is true when H is the bull ( P with anadditional vertex adjacent to the two central vertices). It is easy to see that if the conjecture holdsfor H then it also holds for the complement H . An important result of Alon, Pach and Solymosi[2] shows that if the conjecture holds for H and H ′ then it also holds for the graph obtained bysubstituting H ′ into a vertex of H . It follows that the Erd˝os-Hajnal conjecture holds for every graph H in the closure of { K , P , bull } under complements and substitution. But these are all the graphs(with at least two vertices) for which the conjecture was previously known.The conjecture holds when | H | ≤
4, but is open for three graphs on five vertices: C , P and P . The five-vertex cycle C has been a particularly frustrating open case, and has attracted a gooddeal of unsuccessful attention (for example, it was highlighted by Erd˝os and Hajnal [13] and alsoby Gy´arf´as [16]). So we are happy to report some progress at last: in this paper, we will prove theconjecture for C , and present a number of other results.Let us start by noting that the best known bound for a general graph H is due to Erd˝os andHajnal [13], who showed the following: For every graph H , there exists c > such that max( α ( G ) , ω ( G )) ≥ c √ log | G | for every H -free graph G with | G | ≥ . In an earlier paper [6], with Jacob Fox, we improved on 1.2 for the graph H = C : There exists c > such that max( α ( G ) , ω ( G )) ≥ c √ log | G | log log | G | for every C -free graph G with | G | ≥ . C : There exists τ > such that every C -free graph G satisfies max( α ( G ) , ω ( G )) ≥ | G | τ . The proof of 1.4 is novel, but the same proof method, with some extra twists, yields some otherresults about the Erd˝os-Hajnal conjecture. It does not seem to show that P has the Erd˝os-Hajnalproperty, which, with its complement, is the other open case of 1.1 with | H | = 5; but it does giveother nice things. In particular, it gives results when certain pairs or small families of inducedsubgraphs are excluded.If H is a set of graphs, G is H -free if it is H -free for each H ∈ H . Let H be a set of graphs(or a single graph); we say that H has the Erd˝os-Hajnal property if there exists τ > α ( G ) , ω ( G )) ≥ | G | τ for all H -free graphs. If H = { H } we simply say that H has the Erd˝os-Hajnal property . Thus 1.1 says that every graph has the Erd˝os-Hajnal property, and 1.4 says that C has the Erd˝os-Hajnal property. Note that if H has the Erd˝os-Hajnal property then so does theset { H : H ∈ H} of complements of members of H .There has been some recent progress on small sets of graphs with the Erd˝os-Hajnal property.After partial results by a number of authors (see [4, 5, 17]), the following result was shown in [8]: If F and H are forests then { F, H } has the Erd˝os-Hajnal property. In this paper, we will show that a number of other sets of graphs have the Erd˝os-Hajnal property.For instance, let c C be the graph obtained from a cycle C of length five by adding a new vertex withtwo neighbours in V ( C ), adjacent; and in general, let H denote the complement graph of a graph H . We will show: { c C , c C } has the Erd˝os-Hajnal property. Since c C contains P , this implies that { c C , P } has the Erd˝os-Hajnal property, strengthening thetheorem of [9] that the set of all “hole-with-hat” graphs has the Erd˝os-Hajnal property. It alsoimplies the result of Chudnovsky and Safra [7] that the bull has the Erd˝os-Hajnal property, becauseboth c C , c C contain the bull.We will show that { C , C } has the Erd˝os-Hajnal property. and { C , C } has the Erd˝os-Hajnal property. It would be nice to know if the same is true for { C , C } , but this remains open. We will show thatone of the forests in 1.5 can be replaced by a cycle: If C is a cycle and H is a forest then { C, H } has the Erd˝os-Hajnal property. Some papers say “the class of H -free graphs has the Erd˝os-Hajnal property” in this situation, but here the definitionwe give is more convenient.
2e will also show: If C is a cycle and ℓ is an integer, the set consisting of C and the complements of all cyclesof length at least ℓ has the Erd˝os-Hajnal property. This strengthens the result of Bonamy, Bousquet and Thomass´e [3] that the set consisting of all cyclesof length at least ℓ and their complements has the Erd˝os-Hajnal property (see [10] for a substantialstrengthening of this result). There are some other more complicated results that we will explainlater.There are a number of different ways to phrase the Erd˝os-Hajnal conjecture. Let us define κ ( G ) = α ( G ) ω ( G ). For a set H of graphs, the following are equivalent: • there exists τ > H -free graph G satisfies max( α ( G ) , ω ( G )) ≥ | G | τ ; • there exists τ > H -free graph G contains as an induced subgraph a cographwith at least | G | τ vertices (this was implicitly used by Erd˝os and Hajnal [13]); • there exists τ > H -free graph G contains as an induced subgraph a perfectgraph with at least | G | τ vertices (this is discussed in [16]); • there exists τ > H -free graph G satisfies κ ( G ) ≥ | G | τ .The version using κ is sometimes easier to work with, and we will mostly use it below.An important ingredient in the paper is a lemma about bipartite graphs that we will prove in thenext section. This originates in a powerful lemma that was proved by Tomon [21], and developedfurther by Pach and Tomon [19]. We prove a significant strengthening of Tomon’s result, and use itto prove a key lemma that will be used in the proofs of all our main theorems.The paper is organized as follows. First we prove the strengthening of Tomon’s theorem that weneed, and then apply it to prove our key lemma; then we prove 1.4; then we extend this approachto see what else we can obtain, in particular proving the other theorems mentioned above.Notation throughout is standard. All graphs in this paper are finite and have no loops or paralleledges. We denote by | G | the number of vertices of a graph G . If X ⊆ V ( G ), G [ X ] denotes thesubgraph of G induced on X . We write C k for the cycle of length k , and P k for the path with k vertices. Logarithms are to base two. Let G be a graph. We say that two sets of vertices A, B ⊆ V ( G ) are complete if that are disjointand every element of A is adjacent to every element of B , and anticomplete if they are disjoint andno element of A is adjacent to an element of B . We say that a set H of graphs has the strongErd˝os-Hajnal property if there exists c > H -free graph G with at least twovertices there are sets A, B ⊆ V ( G ) with | A | , | B | ≥ c | G | such that the pair A , B is either completeor anticomplete. It is easy to prove that if H has the strong Erd˝os-Hajnal property then it has theErd˝os-Hajnal property (see [1, 15]). This approach has been used in a number of papers to provethe Erd˝os-Hajnal property for various sets H (see, for example, [3, 4, 5, 8, 10, 17]).If a finite set H of graphs has the strong Erd˝os-Hajnal property then, by considering sparserandom graphs, it is easy to see that it is necessary for H to contain a forest; and similarly it3s necessary for H to contain the complement of a forest (see [8]). It follows from 1.5 that theseconditions are also sufficient, and so 1.5 characterizes finite sets H that have the strong Erd˝os-Hajnal property (infinite sets are a different matter: for example the set of all cycles has the strongErd˝os-Hajnal property, but does not contain a forest).Tomon [21] made the nice observation that there is a similar but weaker property that can alsobe used to prove the Erd˝os-Hajnal property. Suppose that H is a set of graphs and there are c, k > H -free graph G with | G | ≥
2, there is some t = t ( G ) ≥ V ( G )includes t sets of size at least c | G | /t k that are pairwise complete or pairwise anticomplete (note thatthe strong Erd˝os-Hajnal property is the special case where we can always choose t = 2). We recallthat κ ( G ) = α ( G ) ω ( G ); let us write κ ( n ) for the minimum of κ ( G ) over H -free graphs G with n vertices. It follows that κ ( G ) ≥ tκ ( c | G | /t k ), and it is easily checked that this implies that κ ( n ) ≥ n τ for all n , provided τ > H has the strong Erd˝os-Hajnal property.In order to find the required disjoint sets of vertices, Tomon [21] proved a powerful lemma aboutbipartite graphs, which was developed further by Pach and Tomon [19]. We will make use of thesame idea, but will need to prove a significantly stronger form of the lemma.Let G be a graph, and let t, k ≥ t is an integer. We say (( a i , B i ) : 1 ≤ i ≤ t ) is a( t, k ) -comb in G if: • a , . . . , a t ∈ V ( G ) are distinct, and B , . . . , B t are pairwise disjoint subsets of V ( G ) \{ a , . . . , a t } ; • for 1 ≤ i ≤ t , a i is adjacent to every vertex in B i ; • for i, j ∈ { , . . . , t } with i = j , a i has no neighbour in B j ; and • B , . . . , B t all have cardinality at least k .If A, B ⊆ V ( G ) are disjoint and a , . . . , a t ∈ A , and B , . . . , B t ⊆ B , we call this a ( t, k ) -comb in ( A, B ). Our strengthening of Tomon’s lemma [21] is as follows:
Let G be a graph with a bipartition ( A, B ) , such that every vertex in B has a neighbour in A ;and let Γ , ∆ , d > with d < , such that every vertex in A has at most ∆ neighbours in B . Theneither: • for some integer t ≥ , there is a ( t, Γ t − /d ) -comb in ( A, B ) ; or • | B | ≤ d +1 / − (3 / d Γ d ∆ − d . Proof.
We define a partition of B , formed by pairwise disjoint subsets C , C , . . . of B , definedinductively as follows. Let s ≥
1, and suppose that C , . . . , C s − are defined, and every vertex in A has at most (2 / s − ∆ neighbours in D , where D = B \ ( C ∪ · · · ∪ C s − ). Choose a , a , . . . , a k ∈ A with k maximum such that for 1 ≤ i ≤ k , there are at least (2 / s ∆ vertices in D that are adjacentto a i and to none of a , . . . , a i − . Let C s be the set of vertices in D adjacent to one of a , . . . , a k ;then from the maximality of k , every vertex in A has at most (2 / s ∆ neighbours in D \ C s . Thiscompletes the inductive definition of C , C , . . . . Since every vertex in B has a neighbour in A , itfollows that every vertex in B belongs to some C s .(1) For all s ≥ , we may assume that | C s | ≤ d +1 (2 / s − sd − Γ d ∆ − d . s ≥
1, and let a , . . . , a k be as above (that is, chosen with k maximum such that for 1 ≤ i ≤ k ,there are at least (2 / s ∆ vertices in D = B \ ( C ∪ · · · ∪ C s − ) that are adjacent to a i and to none of a , . . . , a i − .) For 1 ≤ i ≤ k let P i be the set of vertices in D that are adjacent to a i and to none of a , . . . , a i − ; thus each | P i | ≥ (2 / s ∆. For 1 ≤ i ≤ k , let Q i be the set of vertices in D \ P i adjacentto a i ; thus every vertex in Q i is adjacent to one of a , . . . , a i − , and | Q i | ≤ (2 / s − ∆ − (2 / s ∆ = (2 / s ∆ / a i has at most (2 / s − ∆ neighbours in D . Inductively, for i = k, k − , . . . , a i is good if at most | P i | / P i are adjacent to a good vertex in { a i +1 , . . . , a k } . (Thus a k is good, if k > { a i : i ∈ I } be the set of all good vertices; we claim that | I | ≥ k/
2. Let Q be the union of the sets Q i ( i ∈ I ); then Q has cardinality at most | I | (2 / s ∆ /
2. If i ∈ { , . . . , k } \ I ,then at least | P i | / ≥ (2 / s ∆ / P i belong to Q ; and so( k − | I | )(2 / s ∆ / ≤ | Q | ≤ | I | (2 / s ∆ / . Consequently | I | ≥ k/
2. For each i ∈ I , let B i be the set of vertices in P i that are not in Q ; then | B i | ≥ (2 / s ∆ /
2, and (( a i , B i ) : i ∈ I ) is an ( | I | , (2 / s ∆ / A, B ). Let t = | I | ; so we mayassume that either t = 0, or (2 / s ∆ / < Γ t − /d (since otherwise the theorem holds); and in eithercase, t < (2Γ(3 / s / ∆) d . Hence k ≤ / s / ∆) d , and | C s | ≤ / s / ∆) d ∆(2 / s − . This proves (1).Now since d <
1, the sum of (2 / s − sd − over all integers s ≥ / d − (2 / − d , and so | B | = | C | + | C | + · · · ≤ d +1 (3 / d − (2 / − d Γ d ∆ − d = 3 d +1 / − (3 / d Γ d ∆ − d . This proves 2.1.
In this section we use 2.1 to prove our key lemma. Given a vertex x ∈ V ( G ), we will apply 2.1 to thebipartite graph of edges between A = N ( x ) and B = V ( G ) \ ( A ∪ { x } ). By 2.1, this will either giveus a large comb (( a i , B i ) : 1 ≤ i ≤ t ), or it will show that A has poor expansion. In the first case, wetry to use the comb either to find H or to find many large sets of vertices that are pairwise completeor pairwise anticomplete; in the second, as there are no edges between G [ A ] and G [ B \ N ( A )] wecan handle them separately. In both cases, it will be helpful if the set { a , . . . , a t } is a stable set: itturns out that we can build this into the key lemma.Let τ >
0. We say that a graph G is τ -critical if κ ( G ) < | G | τ , and κ ( G ′ ) ≥ | G ′ | τ for every inducedsubgraph G ′ of G with G ′ = G . The next result is the key lemma that unlocked all the main resultsin this paper: 5 .1 For all δ, ε > with ε < / , there exists τ > with the following property. Let G be a τ -critical graph, and let X ⊆ V ( G ) with | X | ≥ δ | G | , such that G [ X ] has maximum degree at most εδ | G | . Then there is a ( t, δ | G | / (400 εt )) -comb (( a i , B i ) : 1 ≤ i ≤ t ) of G [ X ] such that t ≥ / (400 ε ) and { a , . . . , a t } is stable, and there is a vertex v ∈ X adjacent to a , . . . , a t and with no neighboursin B ∪ · · · ∪ B t . Proof.
Choose τ with 0 < τ < − /τ δ + (cid:18) ε + 1920 (cid:19) ( εδ ) − τ < . (This is possible since ε < / τ satisfies the theorem.Let G, X be as in the theorem. We may assume that κ ( G ) ≥
2. It follows that 2 < | G | τ , and so | G | > /τ .Let X = X . Inductively, given a set X i − ⊆ X with X i = ∅ , we make the following definitions: • Let v i ∈ X i − have maximum degree in G [ X i − ]. • Let A i be the set of neighbours of v i in G [ X i − ] (possibly A i = ∅ ). • Let C i ⊆ A i be a stable set with | C i | ≥ | A i | τ /ω ( G ). (This exists, since G is τ -critical. Possibly C i = ∅ , but only if A i = ∅ .) • Let X i be the set of vertices in X i − with no neighbour in { v i } ∪ C i . v i C i A i D i X i X i − Figure 1: Figure for 3.1The inductive definition stops when | X i | = ∅ ; let this occur when i = s say. Thus we define a nestedsequence of subsets X = X ⊇ X ⊇ X ⊇ · · · ⊇ X s = ∅ ;and also vertices v i ∈ X i − \ X i and subsets A i , C i ⊆ X i − \ X i for 1 ≤ i ≤ s . Note that there are noedges between { v i } ∪ C i and X j for i < j .For 1 ≤ i ≤ s , let D i be the set of vertices in X i − not in A i ∪ { v i } , and with a neighbour in C i .Let γ = δ/ (400 ε ).(1) We may assume that | D i | ≤
19 ( γ | A i | · | X | /δ ) / for ≤ i < s . From the choice of v i , every vertex in C i has at most | A i | neighbours in D i . By 2.1 applied to6he bipartite graph between C i and D i , replacing Γ , ∆ , d by γ | X | /δ, | A i | , /
2, we deduce that eitherfor some integer t ≥
1, there is a ( t, γ | X | / ( δt ))-comb in ( C i , D i ), or | D i | ≤ / / − (3 / / ( γ | X | · | A i | /δ ) / . Suppose the first holds, and let the comb be (( a j , B j ) : 1 ≤ j ≤ t ). The sets B , . . . , B t are pairwisedisjoint subsets of X , and so tγ | X | / ( δt ) ≤ | X | , that is, t ≥ γ/δ = 1 / (400 ε ). Since | X | ≥ δ | G | , itfollows that γ | X | / ( δt ) ≥ γ | G | /t = δ | G | / (400 ǫt );and therefore in this case the conclusion of the theorem is true. So we may assume that the secondbullet holds. Since 3 / / (3 / − (3 / / ) ≤
19, this proves (1).For 1 ≤ i ≤ s , let x i = | A i | / | X | . Since C ∪ · · · ∪ C s is stable, and hence has cardinality at most α ( G ), and | C i | ≥ | A i | τ ω ( G ) = ( x i | X | ) τ ω ( G ) ≥ ( x i δ | G | ) τ ω ( G ) ≥ ( x i δ ) τ α ( G )for each i , it follows that P ≤ i ≤ s x τi < δ − τ .Now X is partitioned into the sets { v i } (1 ≤ i ≤ s ), A i (1 ≤ i ≤ s ) and D i (1 ≤ i ≤ s ), and so X ≤ i ≤ s (1 + | A i | + | D i | ) = | X | , that is, s | X | + X ≤ i ≤ s | A i || X | + X ≤ i ≤ s | D i || X | = 1 . We will bound these three terms separately.First, since { v , . . . , v s } is stable, it follows that s | X | ≤ α ( G ) | X | ≤ | G | τ | X | ≤ | G | τ − δ , and since | G | τ − ≤ − /τ (because | G | ≥ /τ ), it follows that s/ | X | < − /τ /δ .Second, X ≤ i ≤ s | A i || X | = X ≤ i ≤ s x i = X ≤ i ≤ s x τi x − τi ≤ X ≤ i ≤ s x τi ε − τ ≤ ε ( εδ ) − τ since x i = | A i | / | X | ≤ εδ | G | / | X | ≤ ε .Third, X ≤ i ≤ s | D i || X | ≤ (cid:16) γδ (cid:17) / X ≤ i ≤ s x / i = 1920 ε − / X ≤ i ≤ s x / i by (1) and the definition of γ ; and X ≤ i ≤ s x / i = X ≤ i ≤ s x τi x / − τi ≤ X ≤ i ≤ s x τi ε / − τ ≤ ε / ( εδ ) − τ . X ≤ i ≤ s | D i || X | ≤ εδ ) − τ . Summing, we deduce that 2 − /τ δ + (cid:18) ε + 1920 (cid:19) ( εδ ) − τ ≥ , contrary to the choice of τ . This proves 3.1.This gives us a ( t, δ | G | / (400 εt ))-comb. The t in the denominator comes from applying 2.1with d = 1 /
2; as was observed by Pach and Tomon [19], we could apply 2.1 with d = 1 /k , for anyreal k >
1, and produced a comb with t k in the denominator, but there is no gain for us in theapplications. C In this section we prove 1.4. This is implied by each of several stronger results later in the paper,but since the C result is of great interest, and the argument for C is easier than the materialto come later (which will require additional ideas), we give a separate proof. We will need R¨odl’stheorem [20]: For every graph H and all ε > , there exists δ > such that for every H -free graph G , thereexists X ⊆ V ( G ) with | X | ≥ δ | G | , such that one of G [ X ] , G [ X ] has at most ε | X | ( | X | − edges. We also need the following:
Let G be a graph with at most ε | G | ( | G | − / edges; then for every integer m ≥ with m ≤ ( | G | + 1) / , there exists X ⊆ V ( G ) with | X | = m such that G [ X ] has maximum degree less than ε ( m − . Proof.
By averaging over all subsets Y of V ( G ) with cardinality 2 m −
1, it follows that there existssuch a set Y where G [ Y ] has at most ε (2 m − m − < εm ( m −
1) edges. Thus fewer than m vertices in Y have at least ε ( m −
1) neighbours in Y . Hence there exists X ⊆ Y with | X | = m suchthat G [ X ] has maximum degree less than ε ( m − For every graph H and all ε > , there exists δ > such that for every H -free graph G , thereexists X ⊆ V ( G ) with | X | ≥ δ | G | , such that one of G [ X ] , G [ X ] has maximum degree at most εδ | G | . Proof.
Let ε ′ = ε/
2. By 4.1 there exists δ ′ > H -free graph G , thereexists Z ⊆ V ( G ) with | Z | ≥ δ ′ | G | , such that one of G [ Z ] , G [ Z ] has at most ε ′ | Z | ( | Z | −
1) edges. Let δ = δ ′ /
2; we claim that δ satisfies the theorem. Thus, let G be H -free. By the choice of δ ′ , there exists Z ⊆ V ( G ) with | Z | ≥ δ ′ | G | , such that one of G [ Z ] , G [ Z ] has at most ε ′ | Z | ( | Z | −
1) = ε | Z | ( | Z | − / G by its complement if necessary, we may assume the first. Let m = ⌈ δ | G |⌉ ;then | Z | ≥ ⌈ δ ′ | G |⌉ ≥ m − .
8y 4.2 applied to G [ Z ], there exists X ⊆ Z with | X | = m such that G [ X ] has maximum degree lessthan ε ( m − ≤ εδ | G | . This proves 4.3.If X ⊆ V ( G ), we sometimes write α ( X ) for α ( G [ X ]) and so on. Now we can prove the mainresult of this section, which we restate: C has the Erd˝os-Hajnal property. Proof.
Choose ε with 0 < ε < / δ satisfying 4.3 with H = C . Let τ > τ also satisfies 3.1, and since 400 ε <
1, by reducing τ wemay assume that (400 ε ) − /τ > ε/δ . We will show that κ ( G ) ≥ | G | τ for every C -free graph G . By the remarks in the introduction, this is equivalent to showing that C has the Erd˝os-Hajnalproperty.Suppose that there is a C -free graph G with κ ( G ) < | G | τ , and choose G minimal; then G is τ -critical. By 4.3 there exists X ⊆ V ( G ) with | X | ≥ δ | G | , such that one of G [ X ] , G [ X ] has maximumdegree at most εδ | G | . By replacing G with its complement if necessary (this is legitimate since G is also C -free and τ -critical) we may assume that G [ X ] has maximum degree at most εδ | G | . By3.1 and the choice of τ , there is a ( t, δ | G | / (400 εt ))-comb (( a i , B i ) : 1 ≤ i ≤ t ) of G [ X ] such that t ≥ / (400 ε ) and { a , . . . , a t } is stable, and there is a vertex v ∈ X adjacent to a , . . . , a t and withno neighbours in B ∪ · · · ∪ B t .If there exist i, j with 1 ≤ i < j ≤ t such that some vertex b i ∈ B i has a neighbour b j ∈ B j , thenthe subgraph induced on { b , b , a , a , v } is isomorphic to C , a contradiction. So the sets B , . . . , B t are pairwise anticomplete. Since G is τ -critical, it follows that κ ( B i ) ≥ | B i | τ for each i , and since κ ( B i ) ≤ α ( B i ) ω ( G ), we have α ( B i ) ≥ | B i | τ /ω ( G ) ≥ ( δ | G | / (400 εt )) τ /ω ( G ) . Since B , . . . , B t are pairwise anticomplete, it follows that α ( G ) ≥ X ≤ i ≤ t α ( B i ) ≥ t ( δ | G | / (400 εt )) τ /ω ( G ) , and so κ ( G ) ≥ t ( δ | G | / (400 εt )) τ . Since κ ( G ) < | G | τ , it follows that 400 ε/δ ≥ t /τ − . But t ≥ / (400 ε ), and τ < /
2, and so 400 ε/δ ≥ (400 ε ) − /τ , contrary to the choice of τ . This proves 4.4. Next we will add some refinements to the proof of 1.4, but first let us set up some more terminology.Let G be a graph. A pure pair in G is a pair A, B of disjoint subsets of V ( G ) such that A is eithercomplete or anticomplete to B . A blockade B in G is a sequence ( B , . . . , B t ) of pairwise disjointsubsets of V ( G ) called blocks . (In this paper the order of the blocks B , . . . , B t in the sequence willnot matter.) We denote B ∪ · · · ∪ B t by V ( B ). The length of a blockade is the number of blocks,and its width is the minimum cardinality of a block.A blockade B = ( B , . . . , B t ) in G is pure if ( B i , B j ) is a pure pair for all i, j with 1 ≤ i < j ≤ t .Let P be the graph with vertex set { , . . . , t } , in which i, j are adjacent if B i is complete to B j . We9ay P is the pattern of the pure blockade B . A cograph is a P -free graph. Every cograph P withmore than one vertex admits a pure pair ( A, B ) with
A, B = ∅ and with A ∪ B = V ( P ).We need: Let B = ( B , . . . , B t ) be a pure blockade with a cograph pattern. Then κ ( B ∪ · · · ∪ B t ) ≥ X ≤ i ≤ t κ ( B i ) . Proof.
We proceed by induction on t . If t = 1 the claim is true, so we assume t >
1. Hence there isa partition (
I, J ) of { , . . . , t } , with I, J = ∅ , such that either B i is complete to B j for all i ∈ I and j ∈ J , or B i is anticomplete to B j for all i ∈ I and j ∈ J . We may assume the second by replacing G by its complement if necessary. Let V = V ( B ), and U = S i ∈ I B i , and W = S j ∈ J B j . Thus U isanticomplete to W . From the inductive hypothesis, κ ( U ) ≥ P i ∈ I κ ( B i ) and κ ( W ) ≥ P j ∈ J κ ( B j ).But κ ( V ) = α ( V ) ω ( V ) = ( α ( U ) + α ( W )) ω ( V ) ≥ α ( U ) ω ( U ) + α ( W ) ω ( W ) = κ ( U ) + κ ( W ) , and the result follows. This proves 5.1.The following is a slight extension of an idea of Pach and Tomon [19] (which they called the“quasi-Erd˝os-Hajnal property”): Let τ > , and suppose that G is τ -critical. Then for every integer t > , there is no pureblockade in G with a cograph pattern, of length t and width at least | G | t − /τ , such that B i = V ( G ) for each i . Proof.
Suppose that B = ( B , . . . , B t ) is such a blockade. Since G is τ -critical, κ ( B i ) ≥ | B i | τ ≥| G | τ /t for each i , and so by 5.1, κ ( G ) ≥ κ ( B ∪ · · · ∪ B t ) ≥ X ≤ i ≤ t κ ( B i ) ≥ | G | τ , a contradiction. This proves 5.2. The proof of 4.4 can be developed to give more. We have two ways to do so, and in this section weexplain the first.If H is a graph, we wish to augment it in two ways. Let the vertices of H be { b , . . . , b k } , andadd k + 1 new vertices a , . . . , a k , v to V ( H ), where a i is adjacent to b i for 1 ≤ i ≤ k , and v isadjacent to a , . . . , a k , and there are no other edges. Let the graph we obtain be H ′ . We call H ′ a star-expansion of H .In this section we will prove: 10 .1 Let H be a forest. Let H be the star-expansion of H , and let H be the star-expansion of H .Then { H , H , H , H } has the Erd˝os-Hajnal property. H = P , since P is isomorphic to its complement, and so we onlyhave to exclude two graphs instead of four. We obtain: Let H be the graph of figure 2; then { H, H } has the Erd˝os-Hajnal property. Figure 2: The star-expansion of P .This contains 1.4, 1.7 and 1.8, because the graph of figure 2 contain C , C and C . (The approachvia 6.1 does not work for C , C , because there is no forest H such that the star-expansion of H contains one of C , C .)If B = ( B , . . . , B t ) is a blockade in G , we say an induced subgraph H of G is B -rainbow if V ( H ) ⊆ V ( B ) and | B i ∩ V ( H ) | ≤ ≤ i ≤ t . To prove 6.1 we need the following theorem of [8]: For every forest H , there exist d > and an integer K with the following property. Let G bea graph with a blockade B of length at least K , and let W be the width of B . If every vertex of G has degree less than W/d , and there is no anticomplete pair
A, B ⊆ V ( G ) with | A | , | B | ≥ W/d , thenthere is a B -rainbow copy of H in G . We used this in [8] to deduce that for every forest H , the set { H, H } has the Erd˝os-Hajnalproperty. We see that 6.1 (applied to the forest H ) will be an extension of that result, since the fourgraphs of 6.1 all contain one of H, H .To prove 6.1, we need to bootstrap 6.3 into something stronger, and we do so in several stages.We will use a strengthening of 4.1, due to Nikiforov [18], the following:
For all ε > and every graph H on h vertices, there exist γ, δ > such that if G is a graphcontaining fewer than γ | G | h induced labelled copies of H , then there exists X ⊆ V ( G ) with | X | ≥ δ | G | such that one of G [ X ] , G [ X ] has at most ε | X | ( | X | − edges. Applying 4.2 as before yields:
For all ε > and every graph H on h vertices, there exist γ, δ > such that if G is a graphcontaining fewer than γ | G | h induced labelled copies of H , then there exists X ⊆ V ( G ) with | X | ≥ δ | G | such that one of G [ X ] , G [ X ] has maximum degree at most εδ | G | . For every forest H , there exist d > and an integer K , such that, for every graph G with ablockade B of length at least K , if there is no pure pair A, B ⊆ V ( G ) with | A | , | B | ≥ W/d , where W is the width of B , then there is a B -rainbow copy of one of H, H in G . Proof.
Choose d ′ , K ′ to satisfy 6.3 (with d, K replaced by d ′ , K ′ ). Let ε ≤ / (2 d ′ K ′ ) with ε > γ, δ > K ≥ K ′ /δ , and such that (1 − h/K ) h > − γ . Choose d such that d ≥ d ′ K ′ / ( δK − K ′ ). We claim that K, d satisfy the theorem.(1) δK/K ′ − ≥ max ( εδd ′ K, d ′ /d ).To see that δK/K ′ − ≥ εδd ′ K , observe that δK/ (2 K ′ ) ≥
1, and δK/ (2 K ′ ) ≥ εδd ′ K . The secondpart, that δK/K ′ − ≥ d ′ /d , is true from the choice of d . This proves (1).Let G be a graph with a blockade B = ( B , . . . , B K ) and of width W . We may assume that | B i | = W for each i , and so | V | = KW , where V = B ∪ · · · ∪ B K . We assume that there is no B -rainbow copy of H . But the number of sequences ( v , . . . , v h ) with v , . . . , v h ∈ V , such that v , . . . , v h all belong to different blocks of the blockade, is | V | ( | V | − W )( | V | − W ) · · · ( | V | − ( h − W ) > (1 − h/K ) h | V | h ≥ (1 − γ ) | V | h , and since none of them induce a B -rainbow copy of H , it follows that the number of induced labelledcopies of H in G [ V ] is less than | V | h − (1 − γ ) | V | h = γ | V | h . By 6.5 applied to G [ V ], there exists X ⊆ V with | X | ≥ δKW , such that one of G [ X ] , G [ X ] has maximum degree less than εδKW ; andby replacing G by G if necessary, we may assume that G [ X ] has maximum degree less than εδKW .By (1), there exists a real number W ′ such that δKK ′ − ≥ W ′ W ≥ max (cid:18) εδd ′ K, d ′ d (cid:19) . The sets B ∩ X, . . . , B K ∩ X each have cardinality at most W , but their union has cardinalityat least δKW . Let us choose pairwise disjoint subsets I , . . . , I t of { , . . . , K } , with t maximum suchthat | B ′ h | ≥ W ′ for 1 ≤ h ≤ t , where B ′ h = S i ∈ I h B i ∩ X . We may assume that I , . . . , I t are minimalwith this property, and so | B ′ h | ≤ W ′ + W for 1 ≤ h ≤ t . From the maximality of t , X ( | B i ∩ X | : i ∈ { , . . . , t } \ I ∪ · · · ∪ I t ) < W ′ ;and so δKW ≤ | X | ≤ t ( W ′ + W ) + W ′ . Since δK/K ′ − ≥ W ′ /W it follows that t ≥ K ′ .Let B ′ be the blockade ( B ′ , . . . , B ′ K ′ ); it has width at least W ′ .(2) There is a B ′ -rainbow copy of H . Suppose not. Let G ′ = G [ B ′ ∪ · · · ∪ B ′ K ′ ]. By 6.3 applied to G ′ , either • some vertex of G ′ has degree at least W ′ /d ′ ; or12 there is an anticomplete pair A, B ⊆ V ( G ′ ) with | A | , | B | ≥ W ′ /d ′ .But the first does not hold, since G ′ has maximum degree less than εδKW ≤ W ′ /d ′ ; and the seconddoes not hold, since W ′ /d ′ ≥ W/d and there is no pure pair (
A, B ) in G with | A | , | B | ≥ W/d . Thisproves (2).The B ′ -rainbow copy of H in (2) is also B -rainbow. This proves 6.6. For every forest H , there exist an integer d > , such that, for every integer s ≥ and everygraph G , the following holds. Let D = 2 s − d s − , and let B be a blockade in G of length D . Theneither • G admits a pure blockade A with a cograph pattern, of length s and width at least W/D , where W is the width of B ; or • there is a B -rainbow copy of one of H, H in G . Proof.
Choose
K, d to satisfy 6.6. Then any pair of numbers K ′ , d ′ with K ′ ≥ K and d ′ ≥ d alsosatisfy 6.6, so by increasing K or d if necessary, we may assume that K = d . We claim that d satisfies6.7. This is true if s = 1, from the choice of d , and so we assume it is true for some s ≥ s + 1.Let D = 2 s d s +1 , and let G be a graph with a blockade B = ( B , . . . , B D ) of width W . Partition { , . . . , D } into d sets of cardinality D/d , say I , . . . , I d . Let B ′ h = S i ∈ I h B i for 1 ≤ i ≤ d ; then B ′ = ( B ′ , . . . , B ′ d ) is a blockade of length d and width W D/d . Let G ′ = G [ B ∪ · · · ∪ B D ]. We mayassume there is no B ′ -rainbow copy of H or of H in G ′ ; so from the choice of d , there is a pure pair( A, B ) of G ′ with | A | , | B | ≥ W D/d .Let W ′ = W/ (2 d ), and D ′ = 2 s − d s − . Let p be the number of i ∈ { , . . . , D } such that | A ∩ B i | ≥ W ′ . Then pW + DW ′ ≥ | A | ≥ W D/d , and so p ≥ D/ (2 d ) = D ′ . Let C be theblockade formed by the D ′ largest sets of the form A ∩ B i ; then C has width at least W ′ , and wemay assume that there is no C -rainbow copy of H or of H . Thus the inductive hypothesis, appliedto the blockade C of G [ A ] implies that G [ A ] admits a pure blockade with a cograph pattern, of widthat least W ′ /D ′ = W/D and length 2 s ; and similarly so does G [ B ]. But then combining these givesa pure blockade in G with a cograph pattern, of width at least W/D and length 2 s +1 . This proves6.7.Now finally we can prove 6.1, which we restate: Let H be a forest. Let H be the star-expansion of H , let H be the star-expansion of H , andlet H = { H , H , H , H } . Then H has the Erd˝os-Hajnal property. Proof.
Much of the proof is the same as for 4.4. Let d satisfy 6.7. Choose ε > ε < / (400 d ),choose δ to satisfy 4.3 with H = H , and let γ = δ/ (400 ε ). Choose τ > /τ > ( d ), and such that 2 q d q +1 < / (400 ε ) where q = log ( d ) − log ( γ )1 /τ − − ( d ) . τ sufficiently small we can make q arbitrarily closeto 0, and hence make 2 q d q +1 arbitrarily close to d < / (400 ε ).)As in the proof of 4.4, we may assume that there is a τ -critical H -free graph G , and there exists X ⊆ V ( G ) with | X | ≥ δ | G | , such that G [ X ] has maximum degree at most εδ | G | .By 3.1 and the choice of τ , there is a ( t, γ | G | /t )-comb (( a i , B i ) : 1 ≤ i ≤ t ) of G [ X ] such that t ≥ / (400 ε ) and { a , . . . , a t } is stable, and there is a vertex v ∈ X adjacent to a , . . . , a t and withno neighbours in B ∪ · · · ∪ B t . Let B = ( B , . . . , B t ).(1) There is a B -rainbow copy of H or of H . Suppose not. Choose an integer s maximum such that D s ≤ t , where D s = 2 s − d s − . Thus D s +1 > t . Since D q +1 ≤ / (400 ε ) ≤ t , it follows that s > q .By 6.7, G admits a pure blockade A with a cograph pattern, of width at least γ | G | / ( t D s ) andlength 2 s . By 5.2, γ | G | / ( t D s ) < | G | (2 s ) − /τ , that is, γ < t D s − s/τ . The maximality of s impliesthat 2 s d s +1 ≥ t , and so, substituting for t and for D s , we obtain γ < s d s +2 s − d s − − s/τ . It follows that log ( γ ) + s/τ − s + 1 < (6 s + 1) log ( d ), and so (cid:18) τ − − ( d ) (cid:19) s < log ( d ) − log ( γ ) . Hence s < q , a contradiction. This proves (1).But now the result follows as in 4.4. This proves 6.1. If H is a forest, then since two of the four graphs of 6.1 contain H , it follows that the set consistingof H and the remaining two graphs in 6.1 has the Erd˝os-Hajnal property. But we can do betterthan this: it is sufficient just to exclude one of the remaining two, as we show in this section. Thisis proved by a slight variation in the proof of 6.1.We will need the following theorem of [8] (it is a consequence of 6.3): For every forest H , there exists ε > such that if a graph G with | G | > has maximumdegree less than ε | G | , and has no anticomplete pair of sets A, B ⊆ V ( G ) with | A | , | B | ≥ ε | G | , then G contains H . We use this to prove:
Let H be a forest, and let H ′ be the star-expansion of H . Then H = { H , H ′ } has the Erd˝os-Hajnal property. roof. We define d, ε, δ, τ and the rest, exactly as in the proof of 6.1, except we choose ε satisfying7.1 as well as the other conditions, and choose τ such that εδ > − /τ as well as the other conditions.As before, we may assume that there is a τ -critical H -free graph G , and there exists X ⊆ V ( G )with | X | ≥ δ | G | , such that one of G [ X ] , G [ X ] has maximum degree at most εδ | G | . (We are notfree to replace G by its complement, since the class of H -free graphs is not closed under takingcomplements.)Suppose that G [ X ] has maximum degree at most εδ | G | . By 7.1 applied to G , there exist disjoint A, B ⊆ X , with A complete to B , and with | A | , | B | ≥ εδ | G | . By 5.2, εδ | G | < | G | − /τ , and so εδ < − /τ , contrary to the choice of τ .Thus G [ X ] has maximum degree at most εδ | G | . Exactly as in the proof of 6.1, we obtain theblockade B , and prove there is a B -rainbow copy of H or of H . The second is impossible since G is H -free; and so G contains the star-expansion of H . This proves 7.2.We see that 1.9 follows from 7.2, by applying 7.2 to a forest H ′ containing H and containing apath of length at least | E ( C ) | − H ′ contains C .) The followingis a theorem of Bonamy, Bousquet and Thomass´e [3]: For every integer ℓ > , there exists ε > such that if G has maximum degree less than ε | G | ,and G has no anticomplete pair ( A, B ) with | A | , | B | ≥ ε | G | , then G has a hole of length at least ℓ . The proof of 7.2 can be modified to show the following, by using 7.3 in place of 7.1 (we omit thedetails):
Let H be the star-expansion of a forest; then for every integer ℓ ≥ , { H, C ℓ , C ℓ +1 , C ℓ +2 , . . . } has the Erd˝os-Hajnal property. This implies 1.10, by letting H be the star-expansion of a path of length | E ( C ) | −
4. (We mayassume that C has length at least five, because it is known that C , C both have the Erd˝os-Hajnalproperty.) C with a hat There is still one result mentioned in the introduction that is not contained in any of the results weproved so far, namely 1.6, and now we will prove that. H = { c C , c C } has the Erd˝os-Hajnal property. Proof.
We proceed as usual: as in all these proofs, we choose a suitable ε ≤ /
20, choose δ satisfying4.3, and then choose τ > τ less than any positive function ofthe other parameters we choose. Let us see what we need.We may assume (for a contradiction) that there is a τ -critical H -free graph G ; and there exists X ⊆ V ( G ) with | X | ≥ δ | G | , such that G [ X ] has maximum degree at most εδ | G | . (We can pass to thecomplement if necessary.) Let γ = δ/ (400 ε ). By 3.1, there is a ( t, γ | G | /t )-comb (( a i , B i ) : 1 ≤ i ≤ t )of G [ X ] such that t ≥ / (400 ε ) and { a , . . . , a t } is stable, and there is a vertex v ∈ X adjacent to15 , . . . , a t and with no neighbours in B ∪ · · · ∪ B t .(1) For ≤ i ≤ t , there is a component G [ D i ] of G [ B i ] with | D i | ≥ γ | G | /t . Suppose not, say for i = 1. Choose s maximum such that there are s subgraphs F , . . . , F s of G [ B ], pairwise disjoint, each a union of components of G [ B ], and each with at least γ | G | /t ver-tices. We may assume that each F j is minimal, and so has at most 2 γ | G | /t vertices, since eachcomponent of G [ B ] has at most γ | G | /t vertices. Thus F ∪ · · · ∪ F s has at most 2 sγ | G | /t vertices,and so there are at least γ | G | /t − sγ | G | /t vertices of B not in any of F , . . . , F s . From the max-imality of s , γ | G | /t − sγ | G | /t < γ | G | /t , and so t − s <
1. Hence s ≥ t/
2. But this contradicts5.2, since we will arrange that γ | G | /t ≥ | G | ( t/ − /τ . To ensure this last, arrange at the start ofthe proof that t − τ ≥
4, by choosing 1 / (400 ε ) ≥
16 and τ ≤ /
6, and arrange that γ τ ≥ /
2, bychoosing τ very small. This proves (1).(2) D = ( D , . . . , D t ) is a pure blockade. Suppose not; then there exist distinct i, j ∈ { , . . . , t } , such that some vertex u ∈ D j has botha neighbour and a non-neighbour in D i . Since G [ D i ] is connected, there is an edge xy of G [ D i ]such that u is adjacent to x and not to y ; and then the subgraph induced on { v, a i , a j , x, y, u } isisomorphic to c C , a contradiction. This proves (2).(3) There is no D -rainbow triangle. Suppose there is, and so G contains the star-expansion of K ; but the star-expansion of K contains c C , a contradiction. This proves (3).Let P be the pattern of the pure blockade D . Since P is triangle-free by (3), and | P | = t , itfollows that there is a stable set I of P with cardinality at least t / /
2. Hence the sets D i ( i ∈ I )are pairwise anticomplete, but we will arrange that γ | G | /t ≥ | G | ( t / / − /τ , a contradiction to5.2. To ensure this last, we arrange at the start of the proof that 1 / (400 ε ) ≥
256 and τ ≤ / t ≥
256 and t / − τ ≥
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