Finite Analogs of Szemer\'edi's Theorem

TL;DR

This paper explores finite analogs of Szemerédi's theorem, aiming to deepen understanding and improve bounds on arithmetic progressions, supported by computational tools and theoretical insights.

Contribution

It introduces the theory of finite analogs of Szemerédi's theorem and provides computational implementations to facilitate further research.

Findings

Development of finite analogs of Szemerédi's theorem

Potential for sharpening bounds on arithmetic progressions

Provision of computational tools in Maple, Mathematica, and Java

Abstract

One of the "deepest" theorems in mathematics is Endre Szemer\'edi's theorem about the inevitability of arithmetical progressions. Here we try to nibble at it, by doing "finite" analogs. This is already interesting for its own sake, but we believe that it has the potential to lead to extremely interesting sharpening of the currently rather weak bounds. Let's hope! Along with introducing the theory behind the finite analogs of Szemer\'edi's Theorem, full Maple, Mathematica, and Java code is provided. See paper for further details.