A remarkable lemma of Szemeredi asserts that, very roughly speaking, any dense graph can be decomposed into a bounded number of pseudorandom bipartite graphs. This far-reaching result has proved to… Expand

The existence of sparse graphs having the above Ramsey property as well as the existence of infinitely many critical graphs with respect to the property above are proved.Expand

0. Introduction. In 1936 Erdős and Turan [ET 36] asked whether for every natural number k and every positive constant α, every subset A of [n] = {0, 1, . . . , n − 1} with at least αn elements… Expand

A beautiful result of Szemeredi on the asymptotic structure of graphs is his regularity lemma. Roughly speaking, this result tells us that any large graph may be written as a union of induced, random… Expand

A result is proved that implies that the induced r -size-Ramsey number of the cycle C l is at most c r l for some constant c r that depends only upon r.Expand

The results imply that the evolution of a typical Qn process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices.Expand

This work presents a deterministic algorithm that verifies whether a given m by m bipartite graph G is regular, in the sense of Szemeredi, and makes use of linear-sized expanders to accomplish a suitable form of deterministic sampling.Expand