Paul erdős’ influence on extremal graph theory

Abstract

Paul Erdős is 80 and the mathematical community is celebrating him in various ways. Jarik Nešetřil also organized a small conference in Prague in his honour, where we, combinatorists and number theorists attempted to describe in a limited time the enourmous influence Paul Erdős made on the mathematics of our surrounding (including our mathematics as well). Based on my lecture given there, I shall survey those parts of Extremal Graph Theory that are connected most directly with Paul Erdős’s work. In Turán type extremal problems we usually have some sample graphs L1,…, Lr, and consider a graph G n on n vertices not containing any L i. We ask for the maximum number of edges such a G n can have. We may ask similar questions for hypergraphs, multigraphs and digraphs. We may also ask, how many copies of forbidden subgraphs L i must a graph G n contain with a given number of edges superseding the maximum in the corresponding extremal graph problems. These are the problems on Supersaturated Graphs. We can mix these questions with Ramsey type problems, (Ramsey-Turán Theory). This topic is the subject of a survey by Simonovits and V. T. Sós (Discrete Math 229:293–340, 2001). These topics are definitely among the favourite areas in Paul Erdős’s graph theory.