On one phase transition in the Ramsey theory

Typical results in the classical Ramsey theory assert that in "sufficiently large" combinatorial (and not only) objects there is a simple well-structured subobject. The quantitative analog of this theory asks how big the original object must be to be guaranteed to satisfy this property. The first part of the talk will describe some of the most important results and open problems in this area. The second part will be devoted to joint work with D. Mubayi. It solved, up to one exceptional value, the 1972 Erdős-Hajnal conjecture on the behavior of generalized off-diagonal Ramsey numbers for hypergraphs; all necessary definitions will also be given in the talk.