Phase transitions in the Ramsey-Turán theory

Let f(n) be a function and L be a graph. Denote by RT(n, L, f(n)) the maximum number of edges of an L-free graph on n vertices with independence number less than f(n). Erdős and Sós [1] asked if RT(n,K5, c √ n) = o(n) for some constant c. We answer this question by proving the stronger RT ( n,K5, o (√ n log n )) = o(n). It is known that RT ( n,K5, c √ n log n ) = n/4 + o(n) for c > 1, so one can say that K5 has a Ramsey-Turán phase transition at c √ n log n. We extend this result to several other Ks’s and functions f(n), determining many more phase transitions. We shall formulate several open problems, in particular, whether variants of the Bollobás-Erdős graph exist to give good lower bounds on RT(n,Ks, f(n)) for various pairs of s and f(n). Among others, we use Szemerédi’s Regularity Lemma and the Hypergraph Dependent Random Choice Lemma. We also present a short proof of the fact that Ks-free graphs with small independence number are sparse.