Formulating Szemer\'edi's Theorem in Terms of Ultrafilters

TL;DR

This paper explores Szemerédi's theorem through the lens of ultrafilters and the Stone-Čech compactification, providing a new interpretation and proof framework based on ultrafilter properties.

Contribution

It extends the ultrafilter approach from Van der Waerden's theorem to Szemerédi's theorem, linking combinatorial properties to measure-theoretic ultrafilter characteristics.

Findings

Van der Waerden's theorem is equivalent to the existence of a special ultrafilter.

Szemerédi's theorem corresponds to almost all ultrafilters having certain properties under a counting measure.

Ultrafilter-based interpretation offers a new perspective on arithmetic progressions in dense subsets.

Abstract

Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that form an arithmetic progression. Szemer\'edi's theorem generalizes this statement and asserts that every subset of natural numbers with positive density contains arithmetic progressions of arbitrary length. Van der Waerden's theorem can be proven using elementary combinatorics, but it is also possible to give an interpretation and a short proof in terms of ultrafilters and the Stone-\v{C}ech compactification {\beta}N. This diploma thesis gives an interpretation of Szemer\'edi's theorem in terms of ultrafilters as well. In particular, van der Waerden's theorem is equivalent to the existence of a single ultrafilter with special properties and we will show that Szemer\'edi's theorem is equivalent to the fact that with respect to a counting measure on {\beta}N, almost all ultrafilters have these properties.