2022/07/21 by Balogh, József, Chen, Ce, McCourt, Grace +1 · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2207.10545
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=Ks for s≤ 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 Kr versus a large clique Kn for some r≤ s.