Define a set of lines in $R^3$ to be ``stacked'' with respect to $v \in R^3$ if, from a vantage point far away in the direction of $v$, the lines are linearly ordered by the ``crossing over'' relation. Given a collection of skew lines and a point $v$, we ask, what is the largest stacked subset that must be present among the lines? This question, which appears in a 2000 paper of Erdos, Hajnal and Pach, is intimately related to the well-known Erdos-Hajnal conjecture via the Milnor-Thom theorem. It was recently resolved by a powerful and very general theorem of Alon, Pach, Pinchasi, Radoicic, and Sharir. We describe these results and discuss several related issues, including a generalization to ``Erdos-Hajnal sets'' and an intriguing problem concerning the decomposability of semi-algebraic sets: Do all semi-algebraic sets belong to the set algebra generated by semigroups in $R^d$? Our main result is a resolution of this question in dimensions 1 and 2.
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 $\max(α(G),ω(G))\ge 2^{Ω(\sqrt{\log n \log \log n})}.$
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…
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^{…
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…
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…
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…
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…
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…
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 …
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…
The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $ε>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^ε$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every …
The Ramsey number $r_k(s,n)$ is the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ contains a red set of size $s$ or a blue set of size $n$, where a set is red (blue) if all of its $k$-subsets are red (blue). A $k$-uniform \emph{tight path} of size $s$, denoted by $P_{s}$, is a set of $s$ vertices $v_1 < \cdots < v_{s}$ in $\mathbb{Z}$, and all $s-k+1$ edges of the form $\{v_j,v_{j+1},\ldots, v_{j + k -1}\}$. Let $r_k(P_s, n)$ be the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ results in a red $P_{s}$ or a blue …
In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the…
For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with h(c)=1/2 if 1/4<c<3/4 and h(c)=H(|2c-1|)/2 otherwise, where H is the binary entropy function. Further, we use this result to deduce that almost all graphs in C_n(c) have homogenous sets of linear size. This answers a question raised by Loebl, Reed, Scott, Thomason, and Thomassé [Almost all H-free graphs have the Erdős-Hajnal property] in the case of forbidden i…
