István Tomon
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 $n^{c}$.
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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 $m_H(r)$.
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.
A complement of the Erdős-Hajnal problem on paths with equal-degree endpoints
本研究提出一种处理大等度图的新方法，将Erdős-Hajnal等度端点路径问题的成立范围从n≥600拓展至所有n≥2，完全解决了该问题。
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 Erdős-Hajnal Property for the six-vertex Graph with Edge Set $\{ab,bc,cd,de,af,bf,df\}$
本研究证明了边集为{ab,bc,cd,de,af,bf,df}的6顶点图具有Erdős-Hajnal性质，采用迭代稀疏化方法完成证明。
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.
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.
Pivot-minors and the Erdős-Hajnal conjecture
本文证明了 Kim 和 Oum 的猜想，即每个真轴转子式封闭图类都具有强 Erdős-Hajnal 性质，从而为 Erdős-Hajnal 猜想在轴转子式框架下提供了新的支持。
A problem of Erdős and Hajnal on paths with equal-degree endpoints
本文证明了 Erdős 和 Hajnal 提出的关于图中等度端点路径的问题，给出了边数下界，并推广到偶数阶图及相关极值问题。
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…
An Erdős--Hajnal analogue for permutation classes
针对不含所有分层或共分层排列的排列类，证明其中每个长度为 n 的排列必含长度为 cn 的单调子序列，给出 Erdős--Hajnal 性质的类比结果。
Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures
本文证明了多项式Rödl猜想、多项式Nikiforov猜想与Erdős-Hajnal猜想三者等价，并由此推出字符串图满足多项式Rödl猜想，同时将结果推广至超图、锦标赛、有序图和着色图。
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…
On the solution to the Erdős-Hajnal problem on high-girth high-chromatic subgraphs
本文解释并优化了 Erdős-Hajnal 问题的高围长高色数子图的否定构造，将 Kohlmeyer 和 Kruer 的色数上界从 6 改进到 3。
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 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 …
Induced subgraph density. V. All paths approach Erdos-Hajnal
本文证明 Erdős-Hajnal 猜想对所有路径几乎成立：任意无路径 $H$ 的 $n$ 顶点图都含有规模至少为 $2^{(\log n)^{1-o(1)}}$ 的团或稳定集，将此前仅覆盖至多五顶点路径的结果推广到所有路径。
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…
Erdős-Hajnal conjecture beyond five-vertex graphs
本文证明 Erdős-Hajnal 猜想对不含诱导子图 E-图（在五顶点路径 P5 中间顶点添加一条悬挂边得到的图）的图成立，推广了 Nguyen、Scott 和 Seymour 关于五顶点路径的结果。
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…
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 if $μ>{\rm cf}(μ)=ω$ and $2^μ=μ^+$ then $\binom{μ^+}μ\nrightarrow\binom{μ^+\ \aleph_2}{μ μ}$.
Induced subgraph density. I. A loglog step towards Erdos-Hajnal
本文证明了对于每个图$H$，存在$c>0$使得每个$H$-free图$G$包含大小至少为$2^{c\sqrt{\log |G|\log\log|G|}}$的团或稳定集，改进了Erdős和Hajnal于1977年的经典结果。
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…
Two Erdos-Hajnal-type theorems for forbidden order-size pairs
本文针对超图提出两个Erdős-Hajnal型定理，证明特定条件下超图存在更大齐次集，推进了Axenovich等人提出的相关问题。
The Erdős--Hajnal hypergraph Ramsey problem for $r_4(6,n)$
本文证明了 $r_4(6,n)\ge 2^{2^{cn}}$，改进了先前界，从而确认了所有固定 $k\ge4$ 时 $r_k(k+2,n)$ 的 Erdős--Hajnal 猜想。
Induced subgraph density. IV. New graphs with the Erdős-Hajnal property
本文证明了存在无限多个具有埃尔多斯-哈纳利性质的素图，通过引入迭代稀疏化技术，扩展了有序图和竞赛图的研究。
Towards the Erdős-Hajnal conjecture for $P_5$-free graphs
本文改进了 Erdős-Hajnal 猜想在最小未解决情形（$P_5$-free 图）下的下界，证明了此类图包含大小至少为 $2^{Ω(\log n)^{2/3}}$ 的团或独立集，并将该改进推广至无限族图。
The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every $n$-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $e^{(\log n)^{1 - o(1)}}$. The dependence on the VC-dimension is hidden in the $o(1)$ term. This improves the general lower bound, $e^{c\sqrt{\log n}}$, due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches…
A note on multicolour Erdős-Hajnal conjecture
该研究将多色Erdős-Hajnal猜想归约为主团颜色数与禁用模式相同或多一种的情形，并发现带禁用模式的着色团中齐次集存在非单调性：主图多一种颜色可能减小最大齐次集的大小。
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 …
On a problem of Erdos and Hajnal
研究Erdos和Hajnal关于序数划分关系的问题，在GCH不成立的情况下证明负关系的一致性。
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 $K_{6…
Towards Erdős-Hajnal property for dp-minimal graphs
本文通过引入强$\inom{k}{2}$-自由图，并在稀疏性条件下构造彩虹封锁的降阶方法，为dp-极小图的Erdős-Hajnal性质提供了中间步骤。
Erdős-Hajnal problems for posets
该论文研究偏序集的 Erdős-Hajnal 问题，给出了着色偏序集、反链、链及小布尔格的 Erdős-Hajnal 数的界，并将偏序集 Ramsey 数的下界从 $2n+1$ 改进到 $2.02n$。
A note on Erdős-Hajnal property for graphs with VC dimension $\leq 2$
本文利用现有技术与替换方法，证明了 VC 维度 $\leq 2$ 的图满足 Erdős-Hajnal 性质，并指出 VC 维度 $1$ 图的该性质及可定义对称二元关系下的相关定理可无需 Shelah 的 2-rank 证明。
A parity Erdős-Hajnal theorem for $t$-intersecting curves
对于平面上的$t$-相交曲线族，证明存在大子族使得所有交叉次数奇偶性一致，并应用于拓扑图边数上界。
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…
Rainbow triangles and the Erdős-Hajnal problem in projective geometries
本文提出Erdős-Hajnal猜想的射影几何版本，解决三角形着色所有情形，并证明彩虹三角形自由着色等价于双色分解。
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…
A Single-Exponential Erdős--Hajnal Bound for Graphs of Bounded VC-Dimension
研究有界 VC 维数图的齐次集，改进 Nguyen 等人方法，给出更精确量化界\(\eta_d\geq (Cd)^{-d}\)，通过直接应用 VC 维数假设优化归纳，还推导多种相关图类的定量结果。
A note on strong Erdős-Hajnal for graphs with bounded VC-minimal complexity
受Adler的VC极小理论思路启发，研究者引入VC-极小复杂度概念，证明了有界VC-极小复杂度的有限二部图满足强Erdős-Hajnal性质，即存在正比例大小的二部子集构成完全二部子图或空二部子图。
The following natural problem was raised independently by Erdős-Hajnal and Linial-Rabinovich in the late 80's. How large must the independence number $α(G)$ of a graph $G$ be whose every $m$ vertices contain an independent set of size $r$? In this paper we discuss new methods to attack this problem. The first new approach, based on bounding Ramsey numbers of certain graphs, allows us to improve previously best lower bounds due to Linial-Rabinovich, Erdős-Hajnal and Alon-Sudakov. As an example, we prove that any $n$-vertex graph $G$ having an independent set of size $3$ among every $7$ vertices…
Let $T$ be a tree such that all its vertices of degree more than two lie on one path, that is, $T$ is a caterpillar subdivision. We prove that there exists $ε>0$ such that for every graph $G$ with $|V(G)|\ge 2$ not containing $T$ as an induced subgraph, either some vertex has at least $ε|V(G)|$ neighbours, or there are two disjoint sets of vertices $A,B$, both of cardinality at least $ε|V(G)|$, where there is no edge joining $A$ and $B$. A consequence is: for every caterpillar subdivision $T$, there exists $c>0$ such that for every graph $G$ containing neither of $T$ and its complement as an i…
Induced subgraph density. VII. The five-vertex path
本文证明了关于五顶点路径$P_5$的Erdős-Hajnal猜想，表明无诱导$P_5$的图中存在大规模的 clique 或 stable set。
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