Szemerédi’s Proof of Szemerédi’s Theorem

In 1975, Szemerédi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der Waerden theorem and a lemma now known as the Szemerédi regularity lemma, together with a delicate analysis (based ultimately on double counting arguments) of limiting densities of sets along multidimensional arithmetic progressions. In this note we present an arrangement of this proof that incorporates a number of notational and technical simplifications. Firstly, we replace the use of the regularity lemma by that of the simpler “weak regularity lemma” of Frieze and Kannan. Secondly, we extract the key inductive steps at the core of Szemerédi’s proof (referred to as “Lemma 5”, “Lemma 6”, and “Fact 12” in that paper) as stand-alone theorems that can be stated with less notational setup than in the original proof, in particular involving only (families of) one-dimensional arithmetic progressions, as opposed to multidimensional arithmetic progressions. Thirdly, we abstract the analysis of limiting densities along the (now one-dimensional) arithmetic progressions by introducing the notion of a family of arithmetic progressions with the “double counting property”. We also present a simplified version of the argument that is capable of establishing Roth’s theorem on arithmetic progressions of length three.