Extremal results in random graphs

Abstract

According to Paul Erdős [Some notes on Turán's mathematical work, J. Approx. Theory 29 (1980), page 4] it was Paul Turán who "created the area of extremal problems in graph theory". However, without a doubt, Paul Erdős popularized extremal combinatorics, by his many contributions to the field, his numerous questions and conjectures, and his influence on discrete mathematicians in Hungary and all over the world. In fact, most of the early contributions in this field can be traced back to Paul Erdős, Paul Turán, as well as their collaborators and students. Paul Erdős also established the probabilistic method in discrete mathematics, and in collaboration with Alfréd Rényi, he started the systematic study of random graphs. We shall survey recent developments at the interface of extremal combinatorics and random graph theory.