We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for $k$-uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.
The Erdös-Hajnal conjecture states that for every graph $H$, there exists a constant $δ(H) > 0$ such that every graph $G$ with no induced subgraph isomorphic to $H$ has either a clique or a stable set of size at least $|V(G)|^{δ(H)}$. This paper is a survey of some of the known results on this conjecture.
This paper deals with the minimum number $m_H(r)$ of edges in an $H$-free graph with the chromatic number more than $r$. We show how bounds on Ramsey and Turán numbers imply bounds on $m_H(r)$.
A string graph is the intersection graph of curves in the plane. We prove that there exists an absolute constant $c>0$ such that if $G$ is a string graph on $n$ vertices, then $G$ contains either a clique or an independent set of size at least $n^{c}$.
The Erdos-Hajnal conjecture says that for every graph $H$ there exists $c>0$ such that $\max(α(G),ω(G))\ge n^c$ for every $H$-free graph $G$ with $n$ vertices, and this is still open when $H=C_5$. Until now the best bound known on $\max(α(G),ω(G))$ for $C_5$-free graphs was the general bound of Erdos and Hajnal, that for all $H$, $\max(α(G),ω(G))\ge 2^{Ω(\sqrt{\log n })}$ if $G$ is $H$-free. We improve this when $H=C_5$ to $\max(α(G),ω(G))\ge 2^{Ω(\sqrt{\log n \log \log n})}.$
We prove that for every graph $H$, there exists $\varepsilon>0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $|A|, |B|\ge \varepsilon n$ and $A$ is complete or anticomplete to $B$. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erdős-Hajnal conjecture for vertex-minors.
We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and $A$ is either complete or anticomplete to $B$. This proves the analog of the Erdős-Hajnal conjecture for the class of graphs with no pivot-minor isomorphic to $C_k$.
The Erdos-Hajnal conjecture says that for every graph H there exists c>0 such that every graph G not containing H as an induced subgraph has a clique or stable set of cardinality at least |G|^c. 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 c>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 |G|^c.
Given integers $\ell,n$, the $\ell$th power of the path $P_n$ is the ordered graph $P_n^{\ell}$ with vertex set $v_1<v_2<\cdots < v_n$, and all edges of the form $v_iv_j$ where $|i-j|\le \ell$. The ramsey number $r(P_n^{\ell}, P_n^{\ell})$ is the minimum $N$ such that every 2-coloring of ${[N] \choose 2}$ results in a monochromatic copy of $P_n^{\ell}$. It is well-known that $r(P_n^1, P_n^1)=(n-1)^2+1$. For $\ell>1$, Balko-Cibulka-Král-Kynčl proved that $r(P_n^{\ell}, P_n^{\ell})< c_{\ell}n^{128 \ell}$ and asked for the growth rate for fixed $\ell$. When $\ell=2$, we improve this upper bound b…
We prove that there exists $C>0$ such that $ε(H) \geq \frac{C}{|H|^{5}\log(|H|)}$, where $ε(H)$ is the Erdős-Hajnal coefficient of the tournament $H$, for every prime tournament $H$ for which the celebrated Erdős-Hajnal Conjecture has been proven so far. This is the first polynomial bound on the EH coefficient obtained for all known prime Erdős-Hajnal tournaments, in particular for infinitely many prime tournaments. As a byproduct of our analysis, we answer affirmatively the question whether there exists an infinite family of prime tournaments $H$ with $ε(H)$ lower-bounded by $\frac{1}{\textit…
A "hole-with-hat" in a graph $G$ is an induced subgraph of $G$ that consists of a cycle of length at least four, together with one further vertex that has exactly two neighbours in the cycle, adjacent to each other, and the "house" is the smallest, on five vertices. It is not known whether there exists $ε>0$ such that every graph $G$ containing no house has a clique or stable set of cardinality at least $|G|^ε$; this is one of the three smallest open cases of the Erdős-Hajnal conjecture and has been the subject of much study. We prove that there exists $ε>0$ such that every graph $G$ with no h…
A graph class $\mathcal{G}$ has the strong Erdős-Hajnal property (SEH-property) if there is a constant $c=c(\mathcal{G}) > 0$ such that for every member $G$ of $\mathcal{G}$, either $G$ or its complement has $K_{m, m}$ as a subgraph where $m \geq \left\lfloor c|V(G)|\right\rfloor$. We prove that the class of chordal graphs satisfy SEH-property with constant $c = 2/9$. On the other hand, a strengthening of SEH-property which we call the colorful Erdős-Hajnal property was discussed in geometric settings by Alon et al. (2005) and by Fox et al. (2012). Inspired by their results, we show that for e…
The celebrated Erdös-Hajnal conjecture states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. This conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H$-free $n$-vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. This conjecture is proved for few infinite families of tournaments. In this paper we construct…
Let $m(n,r)$ denote the minimal number of edges in an $n$-uniform hypergraph which is not $r$-colorable. For the broad history of the problem see [RaiSh]. It is known that for a fixed $n$ the sequence \[ \frac{m(n,r)}{r^n} \] has a limit. The only trivial case is $n=2$ in which $m(2,r) = \binom{r+1}{2}$. In this note we focus on the case $n=3$. First, we compare the existing methods in this case and then improve the lower bound.
A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős-Hajnal Conjecture, states that for every tournament $S$ there exists $ε(S)>0$ such that if $T$ is an $n$-vertex tournament that does not contains $S$ as a subtournament, then $T$ contains a transitive subtournament on at least $n^{ε(S)}$ vertices. Let $C_5$ 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 $S=C_5$. Here we prove a strengthening of this result, showing that in every tournament…
A celebrated unresolved conjecture of Erdős and Hajnal states that for every undirected graph $H$ there exists $ε(H)>0$ such that every undirected graph on $n$ vertices that does not contain $H$ as an induced subgraph contains a clique or stable set of size at least $n^{ε(H)}$. The conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ε(H)>0$ such that every $H$-free $n$-vertex tournament $T$ contains a transitive subtournament of order at least $n^{ε(H)}$. We say that a tournament is \textit{prime} if it does not have nontrivial homogeneous sets. So …
Erdös-Hajnal conjecture states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. This conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. This conjecture is known to hold for a few infinite families of tournaments. In this paper we construct two n…
In 1961 Erdős and Hajnal introduced the quantity $m(n)$ as the minimum number of edges in an $n$-uniform hypergraph with chromatic number at least 3. The best known lower and upper bounds for $ m(n) $ are $ c_1 \sqrt{\frac{n}{\ln n}} 2^n$ and $c_2 n^2 2^n$ respectively. The lower bound is due to Radhakrishnan and Srinivasan (see \cite{RS}). A natural generalization for $ m(n) $ is the quantity $ m(n,r) $, which is the minimum number of edges in an $n$-uniform hypergraph with chromatic number at least $r+1$. In this work, we present a new randomized algorithm yielding a bound $ m(n,r) \ge c n^{…
In this short note we provide a relatively simple proof of the Erdős-Hajnal conjecture for families of finite (hyper-)graphs without the $k$-order property. It was originally proved by M. Malliaris and S. Shelah in "Regularity lemmas for stable graphs", Transactions AMS, 366, 2014, 1551-1585.
In 1981, Erdős and Hajnal asked whether the sum of the reciprocals of the odd cycle lengths in a graph with infinite chromatic number is necessarily infinite. Let $\mathcal{C}(G)$ be the set of cycle lengths in a graph $G$ and let $\mathcal{C}_\text{odd}(G)$ be the set of odd numbers in $\mathcal{C}(G)$. We prove that, if $G$ has chromatic number $k$, then $\sum_{\ell\in \mathcal{C}_\text{odd}(G)}1/\ell\geq (1/2-o_k(1))\log k$. This solves Erdős and Hajnal's odd cycle problem, and, furthermore, this bound is asymptotically optimal. In 1984, Erdős asked whether there is some $d$ such that each …
Given integers $2\le t \le k+1 \le n$, let $g_k(t,n)$ be the minimum $N$ such that every red/blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ yields either a $(k+1)$-set containing $t$ red $k$-subsets, or an $n$-set with all of its $k$-subsets blue. Erdős and Hajnal proved in 1972 that for fixed $2\le t \le k$, there are positive constants $c_1$ and $c_2$ such that $$ 2^{c_1 n} < g_k(t, n) < twr_{t-1} (n^{c_2}),$$ where $twr_{t-1}$ is a tower of 2's of height $t-2$. They conjectured that the tower growth rate in the upper bound is correct. Despite decades of work on closely related and s…
We prove that for every nowhere dense class of graphs $\mathcal{C}$, positive integer $d$, and $\varepsilon>0$, the following holds: in every $n$-vertex graph $G$ from $\mathcal{C}$ one can find two disjoint vertex subsets $A,B\subseteq V(G)$ such that $|A|\geq (1/2-\varepsilon)\cdot n$ and $|B|=Ω(n^{1-\varepsilon})$ and either $\operatorname{dist}(a,b)\leq d$ for all $a\in A$ and $b\in B$, or $\operatorname{dist}(a,b)>d$ for all $a\in A$ and $b\in B$. We also show some stronger variants of this statement, including a generalization to the setting of First-Order interpretations of nowhere dens…
The Erdős--Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph $H$, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of $r$-uniform hypergraphs. A theorem of Rödl asserts that if an $n$-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to $0$ or $1$) of size $Ω(n)$. We prove that if a $3$-uniform hypergraph is non-universal then it contains an almost homog…
We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Omega(n^{1/3}log^2 n) which uses at most two colors, and this bound is tight up to a constant factor. This verifies a conjecture of Hajnal which is a case of the multicolor generalization of the well-known Erdős-Hajnal conjecture. We further establish a generalization of this result. For fixed positive integers s and r with s at most r, we determine a constant c_{r,s} such that the following holds. Every r-coloring of the edges of the complete graph on n vertices w…
A seminal result of Hajnal and Szemerédi states that if a graph $G$ with $n$ vertices has minimum degree $δ(G) \ge (r-1)n/r$ for some integer $r \ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples which show optimality of the bound on $δ(G)$ are very structured and, in particular, contain large independent sets. In analogy to the Ramsey-Turán theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results: $\bullet$ For any $r > \ell \ge…
Erdős and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n) vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed color patterns ensures larger monochromatic cliques. Specifically, it claims that for any fixed integer k and any clique K on k vertices edge-colored with two colors, there is a positive constant a such that in any complete n-vertex graph edge-colored with two colors that does not contain a copy of K, there is a monochromatic cli…
We prove that for every k, there exists $c_k>0$ such that every graph G on n vertices not inducing a path $P_k$ and its complement contains a clique or a stable set of size $n^{c_k}$.
An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon et al. states that for every tournament H there exists epsilon(H) > 0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A and B, each of cardinality at least c|T| and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-vertex tourn…
Erdős and Hajnal conjectured that, for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain any induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that for every $k$, there exists $c_k>0$ such that every graph $G$ on $n$ vertices not inducing a cycle of length at least $k$ nor its complement contains a clique or a stable set of size $n^{c_k}$.
A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. The conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains a transitive subtournament of order at least $ n^{ε(H)} $. Both the directed and the undirected versions of the conjecture are known to be…
An equivalent directed version of the celebrated unresolved conjecture of Erdos and Hajnal proposed by Alon, Pack, and Solymosi states that for every tournament H there exists epsilon(H)>0 such that every H-free n-vertex tournament T contains a transitive subtournament of order at least n^(epsilon(H)). A tournament H has the strong EH-property if there exists c > 0 such that for every H-free tournament T with |T| > 1, there exist disjoint vertex subsets A and B, each of cardinality at least |T|n and every vertex of A is adjacent to every vertex of B. Berger et al. proved that the unique five-v…
The celebrated Erdős-Hajnal conjecture states that for every proper hereditary graph class $\mathcal{G}$ there exists a constant $\varepsilon = \varepsilon(\mathcal{G}) > 0$ such that every graph $G \in \mathcal{G}$ contains a clique or an independent set of size $|V(G)|^\varepsilon$. Recently, there has been a growing interest in the symmetrized variant of this conjecture, where one additionally requires $\mathcal{G}$ to be closed under complementation. We show that any hereditary graph class that is closed under complementation and excludes a fixed caterpillar as an induced subgraph satisfie…
An ordered graph is a graph with a linear ordering on its vertex set. We prove that for every positive integer $k$, there exists a constant $c_k>0$ such that any ordered graph $G$ on $n$ vertices with the property that neither $G$ nor its complement contains an induced monotone path of size $k$, has either a clique or an independent set of size at least $n^{c_k}$. This strengthens a result of Bousquet, Lagoutte, and Thomassé, who proved the analogous result for unordered graphs. A key idea of the above paper was to show that any unordered graph on $n$ vertices that does not contain an induced …
A celebrated unresolved conjecture of Erdös and Hajnal states that for every undirected graph $H$ there exists $ ε(H) > 0 $ such that every undirected graph on $ n $ vertices that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $ n^{ε(H)} $. This conjecture has a directed equivalent version stating that for every tournament $H$ there exists $ ε(H) > 0 $ such that every $H-$free $n-$vertex tournament $T$ contains a transitive subtournament of size at least $ n^{ε(H)} $. Recently the conjecture was proved for all six-vertex tournaments, except $K_{6…
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erdős and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]
We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform version $r_k(k+1,n)$. This is the first improvement of the original exponential lower bound for $r_4(5,n)$ implicit in work of Erd\H os and Hajnal from 1972 and also improves the current best known bounds for larger $k$ due to the authors. Second, we prove an upper bound for the hypergraph Erd\H os-Rogers function $f^k_{k+1, k+2}(N)$ that is an iterated $(k-13)…
The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that a H-free graph must contain a complete or empty bipartite graph with parts of polynomial size. We prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is H-free, for any given H, then i…
The Erdős-Hajnal Conjecture states that for every $H$ there exists a constant $ε(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or a stable set of size at least $|V(G)|^{ε(H)}$. The Conjecture is still open. Some time ago its directed version was formulated (see:\cite{alon}). In the directed version graphs are replaced by tournaments, and cliques and stable sets by transitive subtournaments. If the Conjecture is not true then the smallest counterexample is a prime tournament. For a long time the Conjecture was known only for finitely many pri…
The following relaxation of the classical problem of determining Ramsey number of a fixed graph has first been proposed by Erdos, Hajnal and Rado over 50 years ago. Given a graph $G$ and an integer $t \geq 2$ determine the minimum number $N$ such that in any $t$-coloured complete graph on $N$ vertices there is a copy of $G$ using only edges of some $t-1$ colours. We determine the answer precisely when $G$ is a path.
