Ramsey-Turán Problems with small independence numbers

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 \(s\leq 13\). 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 \(r\leq s\).