Ramsey-Tur\'an Problems with small independence numbers

TL;DR

This paper investigates the maximum edges in large graphs avoiding small cliques with limited independence, revealing phase transitions linked to inverse off-diagonal Ramsey numbers.

Contribution

It establishes phase transition phenomena for Ramsey-Turán numbers with small cliques using advanced combinatorial techniques.

Findings

Identifies phase transitions in Ramsey-Turán numbers for cliques up to size 13.

Connects the phase transitions to inverse off-diagonal Ramsey numbers.

Employs Szemerédi's Regularity Lemma and dependent random choice methods.

Abstract

Given a graph $H$ and a function $f(n)$, the Ramsey-Tur\'an 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 $s\leq 13$. By applying Szemer\'edi's Regularity Lemma, the dependent random choice method and some weighted Tur\'an-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 $r\leq s$.