• arithmetic progression
• Banach density
• hypernatural numbers
• Loeb measure
• nonstandard analysis
• Szemerédi's theorem
• ultrafilter

In this thesis we provide a proof of Szemerédi's theorem that simplifies the original combinatorial argument from 1975, following a recent work by R. Jin.
The proof uses the methods of nonstandard analysis.
In the first chapter we construct a model of nonstandard analysis, which consists of
a bounded elementary embedding, called "star map", from the "standard universe" into a larger "nonstandard universe", where one has hypernatural numbers and hyperreal numbers.
The main feature of the considered model is that the nonstandard universe coincides with the standard universe, and thus the star map can be iterated to obtain different levels of infinity.
In the second chapter we prove Szemerédi's theorem by exploiting the properties of the notion of
"strong Banach density", a variant of the classical Banach density in which arithmetic progressions are considered instead of intervals. The proof is based on van der Waerden's theorem and on a version of Szemerédi's regularity lemma. It takes place almost entirely in the second level of infinity, except in one passage where the third level also comes into play.
In the third and final chapter of the thesis, some applications of the theorem are mentioned,
and a possible generalisation of the nonstandard proof to the multi-dimensional case is considered.