The Ramsey number $r_k(s,n)$ is the smallest integer $N$ such that every $N$-vertex $k$-graph contains either a copy of $K_s^{(k)}$ or an independent set of size $n$. Erd\H{o}s and Hajnal conjectured that for every fixed $s>k\ge 4$, one has $r_k(s,n)\ge \operatorname{twr}_{k-1}(\Omega(n))$. This conjecture was independently verified by Mubayi and Suk, and by Conlon, Fox and Sudakov, for $k\ge4$ and $s\ge k+3$. In this paper, we prove that $r_4(6,n)\ge 2^{2^{cn}}$ for some absolute constant $c>0$, improving upon our previous bound. Consequently, we confirm the Erd\H{o}s--Hajnal conjecture for $r_k(k+2,n)$ for all fixed $k\ge4$.
For graphs $H$ and $F$, let $\operatorname{ex}(n,H,F)$ denote the maximum number of copies of $H$ in an $F$-free graph of order $n$. Motivated by the Erd\H{o}s-S\'{o}s theorem, Gerbner and Palmer and, independently, Zhao and Peng conjectured that for every tree $T$ of order $k$ and every $3\le r\le k-1,$ $$\operatornam...
The famous and actively studied problem of Brown--Erd\H{o}s--S\'os from 1973 asks for $f^{(r)}(n;s,k)$, the maximum number of edges in an $r$-graph with $n$ vertices in which no $s$ vertices span $k$ or more edges. In this paper, we concentrate on the case $r=3$ and $s=k+2$, with $k\ge2$ fixed and $n\to\infty$; then it...
Levente Bodnár, Oleg Pikhurko, Shu-Min Sun et al.· 0 citations
For a graph $H$, let $f(n,e,H)$ be the least number of colors in an edge-coloring of some $n$-vertex graph with at least $e$ edges in which every copy of $H$ is rainbow. Burr, Erd\H{o}s, Graham, and S\'os conjectured that $f(n,\lfloor n^2/4\rfloor+1,C_{2k+1})=(1/8+o(1))n^2$ for every fixed $k\ge3$, and Buci\'c, Chen, a...
The Brown--Erd\H{o}s--S\'os problem is a fundamental problem in sparse hypergraph Tur\'an theory. For integers $r,k\ge 2$ and $s\ge r$, the problem asks for the maximum number $f^{(r)}(n;s,k)$ of edges in an $n$-vertex $r$-uniform hypergraph containing no $k$ distinct edges spanning at most $s$ vertices. In 1971, Brown...
For graphs $H$ and $F$, let $ex(n,H,F)$ be the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. We study this problem when $H$ is a clique and $F=T_t$which is a fixed tree on $t$ vertices. The Erd\H{o}s--S\'{o}s conjecture concerns the value of $ex(n,K_2, T_t)$. Gerbner and Palmer proposed a more genera...
For integers $k\ge 1$ and $n\ge k+2$, let $g_k(n)$ be the maximum number of edges in an $n$-vertex graph containing no cycle with a vertex incident with at least $k$ chords. Erd\H{o}s conjectured that $g_k(n)=(k+1)(n-k-1)$ for $n\ge 2k+2$. Lewin found a counterexample. Bollob\'as later conjectured that there exists a f...
Xiao-Zheng Chen, Bo Ning· 0 citations
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