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Ramsey-Tur\'an Problems with small independence numbers
Given a graph and a function , the Ramsey-Tur\'an number
is the maximum number of edges in an -vertex -free graph
with independence number at most . For being a small clique, many
results about are known and we focus our attention on
for . 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 is around the
inverse function of the off-diagonal Ramsey number of versus a large
clique for some .Comment: 20 page