Phase transitions in the Ramsey-Turán theory

Let f ( n ) be a function and H be a graph. Denote by RT ( n , H , f ( n ) ) the maximum number of edges of an H-free graph on n vertices with independence number less than f ( n ) . Erd?s and Sos 12] asked if RT ( n , K 5 , c n ) = o ( n 2 ) for some constant c. We answer this question by proving the stronger RT ( n , K 5 , o ( n log ? n ) ) = o ( n 2 ) . It is known that RT ( n , K 5 , c n log ? n ) = n 2 / 4 + o ( n 2 ) for c 1 , so one can say that K 5 has a Ramsey-Turan phase transition at c n log ? n . We extend this result to several other K s 's and functions f ( n ) , determining many more phase transitions. We formulate several open problems, in particular, whether variants of the Bollobas-Erd?s graph exist to give good lower bounds on RT ( n , K s , f ( n ) ) for various pairs of s and f ( n ) . Among others, we use Szemeredi's Regularity Lemma and the Hypergraph Dependent Random Choice Lemma. We also present a short proof of the fact that K s -free graphs with small independence number are sparse: on n vertices have o ( n 2 ) edges.