The ergodic theoretical proof of Szemerédi’s theorem

Abstract

Introduction. In 1975, E. Szemerédi proved the following theorem conjectured some forty years earlier by Erdös and Turan: THEOREM I. Let A C Z be a subset of the integers of positive upper density, then A contains arbitrarily long arithmetic progressions.

A correspondence principle between (hyper)graph theory and probability theory, and the (hyper)graph removal lemma

“…Theorem 2.2 (Furstenberg Recurrence Theorem, [7], [10]). Let (Ω, B max , P) be a probability space (see Appendix Appendix A. for probabilistic notation).…”

“…The deduction of Theorem 2.1 from Theorem 2.2 proceeds by the Furstenberg Correspondence Principle [7], [10], [8]. Let us give a slightly non-standard exposition of this principle (in particular drawing heavily on the language of probability theory), in order to motivate an analogous principle for graphs and hypergraphs in later sections.…”

“…For every m ≥ 1, let G (m) = (V (m) , E (m) ) be a finite undirected graph, and let P (m) be as in Definition 10 We say "essentially" because there is a slight error term coming from the event that x…”

On one result of J. Bourgain

“…It is easy to see that, under certain conditions on the parameters ı and˛; the restrictions imposed on jI j by (12) are much wider than inequality (7). Thus, according to Theorem 1.3, the Bourgain theorem (Theorem 1.2) is unimprovable.…”

A new proof of the density Hales-Jewett theorem

“…A second reason that a new proof of the density Hales-Jewett theorem is interesting is that it immediately implies Szemerédi's theorem, and finding a new proof of Szemerédi's theorem seems always to be illuminating-or at least this has been the case for the four main approaches discovered so far (combinatorial [Sze75], ergodic [Fur77], [FKO82], Fourier [Gow01], hypergraph removal [Gow06], [Gow07], [RS04], [NRS06]). Surprisingly, in view of the fact that DHJ is considerably more general than Szemerédi's theorem and the ergodic-theory proof of DHJ is considerably more complicated than the ergodictheory proof of Szemerédi's theorem, the new proof we have discovered gives arguably the simplest proof yet known of Szemerédi's theorem.…”