Ramsey-Turán problems with small independence numbers

Given a graph H and a function f(n), the Ramsey-Turan 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 <= 13. By applying Szemeredi's Regularity Lemma, the dependent random choice method and some weighted Turan-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 Kr versus a large clique K-n for some r <= s.(c) 2023 Published by Elsevier Ltd.

Automated summary

This article studies the case of small cliques in the Ramsey-Turan number RT(n, H, f(n)), and proves that these cliques have phase transitions under certain conditions using mathematical methods.