# Category Archives: Ramsey Theory

## Optimal intersectivity

In ergodic Ramsey theory, one often wants to prove that certain dynamically defined sets in a probability space intersect (or “recur”) in non-trivial ways. Typically, this is achieved by studying the long term behavior of the sets as the dynamics … Continue reading

## An arithmetic van der Corput trick and the polynomial van der Waerden theorem

The van der Corput difference theorem (or trick) was develop (unsurprisingly) by van der Corput, and deals with uniform distribution of sequences in the torus. Theorem 1 (van der Corput trick) Let be a sequence in a torus . If … Continue reading

## Measure preserving actions of affine semigroups and {x+y,xy} patterns

Vitaly Bergelson and I have recently submitted to the arXiv our paper entitled `Measure preserving actions of affine semigroups and patterns’. The main purpose of this paper is to extend the results of our previous paper, establishing some partial progress … Continue reading

## Szemerédi Theorem Part VI – Dichotomy between weak mixing and compact extension

This is the sixth and final post in a series about Szemerédi’s theorem. In this post I complete the proof of the Multiple Recurrence Theorem, which I showed in a previous post of this series to be equivalent to Szemerédi’s … Continue reading

## New polynomial and multidimensional extensions of classical partition results

Vitaly Bergelson, John Johnson and I recently uploaded to the arXiv a paper entitled “New polynomial and multidimensional extensions of classical partition results“. In this post I will give some motivating examples for the results in the paper. To keep … Continue reading

## A proof of Deuber’s theorem using Hales-Jewett’s theorem

In my previous post I explained how Rado’s theorem follows from Deuber’s theorem (which in turn gives a little more than Rado’s theorem, in one direction). The main purpose of this post is to give a full proof of Deuber’s … Continue reading

## Rado’s theorem and Deuber’s theorem

In this post I talk about (and prove) a fundamental theorem of Rado in Ramsey’s theory. I will prove “half” of the theorem and will postpone the second part of the proof to a future post. To better appreciate Rado’s … Continue reading