Strengthening Rodl's theorem

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rodl showed in the 1980s that every H-free graph has large parts that are very dense or very sparse. More precisely, let us say that a graph F on n vertices is c-restricted if either F or its complement has maximum degree at most cn. Rodl proved that for every graph H, and every c>0, every H-free graph G has a linear-sized set of vertices inducing a c-restricted graph. We strengthen Rodl's result as follows: for every graph H, and all c>0, every H-free graph can be partitioned into a bounded number of subsets inducing c-restricted graphs

A further extension of R\"odl's theorem

Fix $\varepsilon>0$ and a nonnull graph $H$. A well-known theorem of R\"odl from the 80s says that every graph $G$ with no induced copy of $H$ contains a linear-sized $\varepsilon$-restricted set $S\subseteq V(G)$, which means $S$ induces a subgraph with maximum degree at most $\varepsilon\vert S\vert$ in $G$ or its complement. There are two extensions of this result: $\bullet$ quantitatively, Nikiforov (and later Fox and Sudakov) relaxed the condition "no induced copy of $H$" into "at most $\kappa\vert G\vert^{\vert H\vert}$ induced copies of $H$ for some $\kappa>0$ depending on $H$ and $\varepsilon$"; and $\bullet$ qualitatively, Chudnovsky, Scott, Seymour, and Spirkl recently showed that there exists $N>0$ depending on $H$ and $\varepsilon$ such that $G$ is $(N,\varepsilon)$-restricted, which means $V(G)$ has a partition into at most $N$ subsets that are $\varepsilon$-restricted. A natural common generalization of these two asserts that every graph $G$ with at most $\kappa\vert G\vert^{\vert H\vert}$ induced copies of $H$ is $(N,\varepsilon)$-restricted for some $\kappa,N>0$ depending on $H$ and $\varepsilon$. This is unfortunately false, but we prove that for every $\varepsilon>0$, $\kappa$ and $N$ still exist so that for every $d\ge0$, every graph with at most $\kappa d^{\vert H\vert}$ induced copies of $H$ has an $(N,\varepsilon)$-restricted induced subgraph on at least $\vert G\vert-d$ vertices. This unifies the two aforementioned theorems, and is optimal up to $\kappa$ and $N$ for every value of $d$.Comment: 11 pages, revised according to the referees' comment

Induced subgraph density. VI. Bounded VC-dimension

We confirm a conjecture of Fox, Pach, and Suk, that for every $d>0$, there exists $c>0$ such that every $n$-vertex graph of VC-dimension at most $d$ has a clique or stable set of size at least $n^c$. This implies that, in the language of model theory, every graph definable in NIP structures has a clique or anti-clique of polynomial size, settling a conjecture of Chernikov, Starchenko, and Thomas. Our result also implies that every two-colourable tournament satisfies the tournament version of the Erd\H{o}s-Hajnal conjecture, which completes the verification of the conjecture for six-vertex tournaments. The result extends to uniform hypergraphs of bounded VC-dimension as well. The proof method uses the ultra-strong regularity lemma for graphs of bounded VC-dimension proved by Lov\'asz and Szegedy and the method of iterative sparsification introduced by the authors in an earlier paper.Comment: 11 pages, minor revision

Density theorems for bipartite graphs and related Ramsey-type results

In this paper, we present several density-type theorems which show how to find a copy of a sparse bipartite graph in a graph of positive density. Our results imply several new bounds for classical problems in graph Ramsey theory and improve and generalize earlier results of various researchers. The proofs combine probabilistic arguments with some combinatorial ideas. In addition, these techniques can be used to study properties of graphs with a forbidden induced subgraph, edge intersection patterns in topological graphs, and to obtain several other Ramsey-type statements

Induced Ramsey-type theorems

We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.Comment: 30 page

Induced subgraphs density. IV. New graphs with the Erd\H{o}s-Hajnal property

Erd\H{o}s and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ ("$H$-free" means with no induced subgraph isomorphic to $H$). Alon, Pach, and Solymosi reduced the Erd\H{o}s-Hajnal conjecture to the case when $H$ is prime (that is, $H$ cannot be obtained by vertex-substitution from smaller graphs); but until now, it was not shown for any prime graph with more than five vertices. We will provide infinitely many prime graphs that satisfy the conjecture. Let $H$ be a graph with the property that for every prime induced subgraph $G'$ with $|G'|\ge 3$, $G'$ has a vertex of degree one and a vertex of degree $|G'|-2$. We will prove that every graph $H$ with this property satisfies the Erd\H{o}s-Hajnal conjecture, and infinitely many graphs with this property are prime. Our proof method also extends to ordered graphs; and we obtain a theorem which significantly extends a recent result of Pach and Tomon about excluding monotone paths. Indeed, we prove a stronger result, that we can weaken the "$H$-free" hypothesis of the Erd\H{o}s-Hajnal conjecture to one saying that there are not many copies of $H$; and strengthen its conclusion, deducing a "polynomial" version of R\"odl's theorem conjectured by Fox and Sudakov. We also obtain infinitely many new prime tournaments that satisfy the Erd\H{o}s-Hajnal conjecture (in tournament form). Say a tournament is buildable if it can be grown from nothing by repeatedly either adding a vertex of out-degree $\le 1$ or in-degree $\le 1$, or vertex-substitution. All buildable tournaments satisfy the tournament version of the Erd\H{o}s-Hajnal conjecture.Comment: 19 page

Induced subgraph density. III. The pentagon and the bull

A theorem of R\"odl says that for every graph $H$, and every $\varepsilon>0$, there exists $\delta>0$ such that if $G$ is a graph that has no induced subgraph isomorphic to $H$, then there exists $X\subseteq V(G)$ with $|X|\ge \delta|G|$ such that one of $G[X],\overline{G}[X]$ has at most $\varepsilon\binom{|X|}{2}$ edges. But for fixed $H$, how does $\delta$ depends on $\varepsilon$? If the dependence is polynomial, then $H$ satisfies the Erd\H{o}s-Hajnal conjecture; and Fox and Sudakov conjectured that the dependence is polynomial for {\em every} graph $H$. This conjecture is substantially stronger than the Erd\H{o}s-Hajnal conjecture itself, and until recently it was not known to be true for any non-trivial graphs $H$. The preceding paper of this series showed that it is true for $P_4$, and all graphs obtainable from $P_4$ by vertex-substitution. Here we will show that the Fox-Sudakov conjecture is true for all the graphs $H$ that are currently known to satisfy the Erd\H{o}s-Hajnal conjecture. In other words, we will show that it is true for the bull, and the 5-cycle, and induced subgraphs of them, and all graphs that can be obtained from these by vertex-substitution. There is a strengthening of R\"odl's theorem due to Nikiforov, that replaces the hypothesis that $G$ has no induced subgraph isomorphic to $H$, with the weaker hypothesis that the density of induced copies of $H$ in $G$ is small. We will prove the corresponding ``polynomial'' strengthening of Nikiforov's theorem for the same class of graphs $H$

Erdős–Hajnal for graphs with no 5-hole

© 2023 The Authors. Proceedings of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.The Erdős–Hajnal conjecture says that for every graph H there exists t > 0 such that every graph G not containing H as an induced subgraph has a clique or stable set of cardinality at least IGIt. 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 complement of a forest, there exists t > 0 such that every graph G containing neither of C,H as an induced subgraph has a clique or stable set of cardinality at least IGIt.EPSRC, EP/V007327/1 || AFOSR, A9550-19-1-0187, FA9550-22-1-0234 || National Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2020-03912 || NSF, DMS-1763817, DMS-215416

Edge distribution of graphs with few copies of a given graph

We show that if a graph contains few copies of a given graph, then its edges are distributed rather unevenly. In particular, for all ε \u3e 0 and r ≥ 2, there exist ζ= ζ(ε,r) \u3e 0 and k = k(ε,r) such that, if n is sufficiently large and G = G(n) is a graph with fewer than ζnr r-cliques, then there exists a partition V(G) = ⊂i=0k Vi such that Vi = ⌊ n/k⌋ and e(Wi)\u3c εVi2 for every i ∈ [k]. We deduce the following slightly stronger form of a conjecture of Erdos. For all c \u3e 0 and r \u3e 3, there exist ζ= ζ(c,r) \u3e 0 and β= β(c,r) \u3e 0 such that, if n is sufficiently large and G = G(n, [cn2]) is a graph with fewer than ζnr r-cliques, then there exists a partition V(G) = V1 ⊂ V2 with |V 1|= ⌊n/2⌋ and V2 = ⌈n/2⌉ such that e(V1,V2) \u3e (1/2 + β)e(G)