Combinatorial Aspects of Szemerédi ’ S Theorem

In the first section we review the beautiful combinatorial theory of Ramsey as well as the history of Szemerédi’s theorem. In Section 2 we give a sketch of Ruzsa’s modern proof of Freiman’s theorem on sumsets which plays an important role in Gowers’ quantitative proof of Szemerédi’s theorem. In the last section we deduce Roth’s theorem (Szemerédi’s theorem in the case k=3) from the Triangle Removal Lemma (in graph-theoretic language) which is an application of Szemerédi’s Regularity Lemma (a powerful tool in graph theory), and show the BalogSzemerédi-Gowers theorem by a graph-theoretic method.