Recent Developments in Extremal Combinatorics: Ramsey and Turan Type Problems

Extremal combinatorics is one of the central branches of discrete mathematics and has experienced an impressive growth during the last few decades. It deals with the problem of determining or estimating the maximum or minimum possible size of a combinatorial structure which satisfies certain requirements. Often such problems are related to other areas including theoretical computer science, geometry, information theory, harmonic analysis and number theory. In this paper we discuss some recent advances in this subject, focusing on two topics which played an important role in the development of extremal combinatorics: Ramsey and Turan type questions for graphs and hypergraphs.