Introduction

What does a typical triangle-free graph look like? This question was answered by Erdős, Kleitman and Rothschild proving that almost every triangle-free graph is bipartite, i.e., is two-colorable.

From the point of view of extremal graph theory, this result resembles an old result of Mantel stating that the complete bipartite graph is the extremal, i.e., edge-maximum, triangle-free graph. Mantel's solution was a kind of forerunner of extremal graph theory. Its starting point is usually considered to be Turán's celebrated generalization of Mantel's result, characterizing the extremal graphs Tl(n) on n vertices which do not contain a complete graph Kl + 1 on l + 1 vertices as a subgraph. Turán's result stimulated a variety of deep results in graph theory, the reader is referred to and, two excellent sources on these problems. For our purposes we will just mention two strengthenings of Turán's theorem.

Let H be a graph of chromatic number l + 1. By orbn(H) we denote the class of all graphs on n vertices that do not contain H as a weak subgraph, i.e., the class of all H-free graphs.