A Simple Combinatorial Proof of Szemer\'{e}di's Theorem via Three Levels of Infinities

TL;DR

This paper offers a straightforward nonstandard proof of Szemerédi's theorem using three levels of infinities and elementary embeddings, simplifying the traditional complex combinatorial arguments.

Contribution

It introduces a novel nonstandard approach employing three levels of infinities and elementary embeddings to prove Szemerédi's theorem more simply.

Findings

Proof leverages nonstandard analysis techniques

Simplifies the combinatorial proof of Szemerédi's theorem

Uses three levels of infinities and elementary embeddings

Abstract

We present a nonstandard simple elementary proof of Szemer\'{e}di's theorem by a straightforward induction with the help of three levels of infinities and four different elementary embeddings in a nonstandard universe.