Ramsey-Turán Problems with small independence numbers

Abstract

Given a graph $H$ and a function $f(n)$, the Ramsey-Turán number $RT(n,H,f(n))$ is the maximum number of edges in an $n$-vertex $H$-free graph with independence number at most $f(n)$. For $H$ being a small clique, many results about $RT(n,H,f(n))$ are known and we focus our attention on $H=K_s$ for $s\le 13$. By applying Szemerédi's Regularity Lemma, the dependent random choice method and some weighted Turán-type results, we prove that these cliques have the so-called phase transitions when $f(n)$ is around the inverse function of the off-diagonal Ramsey number of $K_r$ versus a large clique $K_n$ for some $r\le s$.

20 pages