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). Erdos and Sós  asked if RT for some constant c. We answer this question by proving the stronger RT. It is known that RT for c>1, so one can say that K5 has a Ramsey-Turán phase transition at cnlogn. We extend this result to several other Ks's and functions f(n), determining many more phase transitions. We formulate several open problems, in particular, whether variants of the Bollobás-Erdos 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: on n vertices have o(n2) edges.
- Dependent random choice
- Independence number
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics