Holes in Graphs

The celebrated Regularity Lemma of Szemerédi asserts that every sufficiently large graph G can be partitioned in such a way that most pairs of the partition sets span -regular subgraphs. In applications, however, the graph G has to be dense and the partition sets are typically very small. If only one -regular pair is needed, a much bigger one can be found, even if the original graph is sparse. In this paper we show that every graph with density d contains a large, relatively dense -regular pair. We mainly focus on a related concept of an ( , σ)-dense pair, for which our bound is, up to a constant, best possible.