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=Ks
for s≤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 Kr versus a large
clique Kn for some r≤s.Comment: 20 page