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Tournaments and the Strong Erd\H {O} S-Hajnal Property

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Tournaments and the Strong Erd\H {O} S-Hajnal Property TOURNAMENTS AND THE STRONG ERDOS-HAJNAL} PROPERTY ELI BERGER, KRZYSZTOF CHOROMANSKI, MARIA CHUDNOVSKY, AND SHIRA ZERBIB Abstract. A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erd}os-Hajnal Conjecture, states that for every tournament S there exists "(S) > 0 such that if T is an n-vertex tournament that does not contains S as a subtour- nament, then T contains a transitive subtournament on at least "(S) n vertices. Let C5 be the unique five-vertex tournament where every vertex has two inneighbors and two outneighbors. The Alon- Pach-Solymosi conjecture is known to be true for the case when S = C5. Here we prove a strengthening of this result, showing that in every tournament T with no subtorunament isomorphic to C5 there exist disjoint vertex subsets A and B, each containing a linear proportion of the vertices of T , and such that every vertex of A is adjacent to every vertex of B. 1. introduction A tournament is a complete graph with directions on edges. A tour- nament is transitive if it has no directed triangles. For tournaments S; T we say that T is S-free if no subtournament of T is isomorphic to S. In [1] a conjecture was made concerning tournaments with a fixed forbidden subtournament: Conjecture 1.1. For every tournament S there exists " > 0 such that every S-free n-vertex tournament contains a transitive subtournament on at least n" vertices. arXiv:2002.07248v2 [math.CO] 14 Sep 2021 It was shown in [1] that Conjecture 1.1 is equivalent to the Erd}os- Hajnal Conjecture [7, 8]. Conjecture 1.1 is known to hold for a few types of tournaments S [2, 3], but is still wide open in general. Maria Chudnovsky was supported by NSF grant DMS-1763817. This material is based upon work supported in part by the U. S. Army Research Laboratory and the U. S. Army Research Office under grant number W911NF1610404. Shira Zerbib was supported by NSF grant DMS-1953929. Eli Berger, Maria Chudnovsky and Shira Zerbib were supported by US-Israel BSF grant 2016077. 1 5 The tournament C5 In this section we prove 1.5. We start with some preliminary observations. Let v1,...,v5 be the vertices of C5. Then there exists an ordering (v✓(1),v✓(2),v✓(3),v✓(4),v✓(5)) of v1,...,v5 where the set of backward edges is the following: (v ,v ), (v ,v ), (v ,v ) . We call this ordering the { ✓(5) ✓(1) ✓(4) ✓(1) ✓(5) ✓(2) } path ordering of C5 since under this ordering the set of backward edges forms a path (and one isolated vertex). There also exists an ordering (v⇢(1),v⇢(2),v⇢(3),v⇢(4),v⇢(5)) of the vertices of C5 where the set of backward edges is (v ,v ), (v ,v ), (v ,v ), (v ,v ) . We call this ordering the { ⇢(5) ⇢(3) ⇢(3) ⇢(1) ⇢(5) ⇢(1) ⇢(4) ⇢(2) } cyclic ordering of C5, since under this ordering the set of backward edges forms a graph containing a cycle (a triangle plus an edge). 1 5.1 Let c, d > 0, 0 < λ < 1, ✏ < log dc ( ) and w =(0, 0, 1, 0, 0). Let (S1,...,S5) be a (c, λ,w)- 2 2 structure of an ✏-critical tournament T . Let s S ,s S ,s S .Assumethats is adjacent to 1 2 1 3 2 3 5 2 5 5 both s1 and s3 and s3 is adjacent to s1. Let Sˆ2 be the subset of the vertices of S2 adjacent to s3,s5 and from s1. Let Sˆ4 be the subset of the vertices of S4 adjacent to s5 and from s1,s3.Assumethat Sˆ d S for i 2, 4 . Then T contains a copy of C . | i| ≥ | i| 2 { } 5 1 ˆ ˆ Proof. By 2.7, and since T is ✏-critical and ✏ < log dc ( 2 ),thereexists2 S2 and s4 S4 such 2 2 2 that s is adjacent to s .Butnow s ,...,s induces a copy of C in T and the ordering (s ,...,s ) 4 2 { 1 5} 5 1 5 is a cyclic ordering. We will now prove 1.5 which we restate below: 2 E. BERGER, K. CHOROMANSKI, M. CHUDNOVSKY, AND S. ZERBIB 5.2 The tournament C5 satisfies the Erd˝os-Hajnal Conjecture. 2 3 1 4 5 Fig.3 Tournament C5 - the smallest tournament that is not a galaxy. Figure 1. The C5 tournament. Proof. Assume otherwise. Taking ✏ > 0 small enough, we may assume that there exists a C5-free Let D be a directed graph, and let A; B V (D) with A B = . 1 ✏-critical tournament T . By 2.11 T contains a (c, λ,w)-structure⊆ (S1,...,S5) for\ some;c>0, λ = 720 and w =(0, 0, 1We, 0, 0). say We that mayA is assumecomplete without to B lossif all of every generality vertex that of ASis adjacentmod 3 = to 0. | 3| Let (T ,T ,Tevery)bea(3 vertex, 1 of)-subdivisionB, and that ofASis.complete Let M = from 30. LetB ifS everybe the vertex subset of A ofisS of M-good 1 2 adjacent3 from3 every vertex of 3. A class of tournamentsi⇤ is hereditary if i vertices with respect to (S ,...,S ). By 2.14B we have S (1 5 ) S . Denote T = S T it is closed under1 subtournaments.5 A hereditary| i⇤| ≥ class− M of| tournamentsi| i⇤ 3⇤ \ i has the strong Erd}os-Hajnalproperty if there exists " = "( ) such thatT for every T there exist disjoint12 subsets A, B of V (T ),T each of size " V (T ) such2 that T A is complete to B. Thej followingj question is closely related to Conjecture 1.1. Question 1.1. For which tournaments S does the class of S-free tour- naments have the strong Erd¨os-Hajnalproperty? It is easy to see [2] that if S is a tournament, and the class of S-free graphs has the strong Erd¨os-Hajnal property, then 1.1 is true for S. In [4] there is a list of necessary conditions for a tournament S to satisfy 1.1. Denote by C5 the (unique) tournament on 5 vertices in which every vertex is adjacent to exactly two other vertices. One way to construct this tournament is with vertex set 0; 1; 2; 3; 4 and i is adjacent to i+1 mod 5 and i + 2 mod 5 (see Figuref 1). g In [2] it was proved that C5 satisfies the Erd}os-Hajnal conjecture. Here we prove the following stronger result: Theorem 1.2. The class of C5-free tournaments has the strong Erd}os- Hajnal property. TOURNAMENTS AND THE STRONG ERDOS-HAJNAL} PROPERTY 3 2. Regularity Tools We recall some definitions given in [3]. Let c > 0, 0 < λ < 1 be constants, and let w be a 0; 1 -vector of length w . Let T be a tournament with V (T ) = n. Denotef g by tr(T ) the largestj j size of the transitive subtournamentj j of T . For " > 0 we call a tournament T"-critical for " > 0 if tr(T ) < T " but for every proper subtournament S of T we have: tr(S) S "j. Aj sequence of disjoint > j j subsets (S1;S2; :::; S w ) of V (T ) is a (c; λ, w)-structure if j j whenever wi = 0 we have Si > cn, • whenever w = 1 the subtournamentj j induced on S is transitive • i i and Si > c tr(T ), d+(Sj ;Sj ) ·1 λ for all 1 i < j w . • i j > − 6 6 j j We say that a (c; λ, w)-structure is smooth if the last condition of the definition of the (c; λ, w)-structure is satisfied in a stronger form, namely we have: for every i < j, every v Si has at most λ Sj inneighbors in S , and every v S has at most2 λ S outneighborsj inj j 2 j j ij Si. Theorem 3.5 of [3] asserts: Theorem 2.1. Let S be a tournament, let w be a 0; 1 -vector, and 1 f g let 0 < λ < 2 be a constant. Then there exist "; c > 0 such that every S-free "-critical tournament contains a smooth (c; λ, w)-structure. Here we need a weaker form of Theorem 2.1 for the case when w is the all-zero vector. It turns out that in that case we do need the criticality assumption. The proof consists of standard regularity lemma arguments, and can be easily reconstructed from the proof of 2.1 and 2.8 in [2]. Thus we have: Theorem 2.2. Let S be a k-vertex tournament and let w be an all-zero vector. There exists c > 0 such that every S-free tournament contains 1 a smooth (c; k ; w)-structure. 3. A lemma on outsimplicial directed graphs We say that a directed graph is outsimplicial if the outneighborhood of each vertex is a clique in the underlying undirected graph. Next we prove the main lemma that we use for the proof of Theorem 1.2. The ideas in the proof of Lemma 3.1 seem to go beyond the particular setup of the lemma. In fact, they were later used in [5] to prove the analogue of Theorem 1.2 to three 6-vertex tournaments containing C5. 4 E. BERGER, K. CHOROMANSKI, M. CHUDNOVSKY, AND S. ZERBIB Lemma 3.1. If D is an outsimplicial directed graph on n > 1 vertices, then there exist two disjoint subsets A and B of V (D), both of size at least n=6 , such that either b c (i) there is no edge, in any direction, between a vertex of A and a vertex of B, or (ii) there is a path from every vertex of A to every vertex of B.
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