  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 $r\leq s$.
