
Recent Posts
 An arithmetic van der Corput trick and the polynomial van der Waerden theorem
 Piecewise syndetic sets, topological dynamics and ultrafilters
 Measure preserving actions of affine semigroups and {x+y,xy} patterns
 Szemerédi Theorem Part VI – Dichotomy between weak mixing and compact extension
 Gaussian systems
Tag Archives: van der waerden
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
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
HalesJewett and a generalized van der Warden Theorems
The HalesJewett theorem is one of the most fundamental results in Ramsey theory, implying the celebrated van der Waerden theorem on arithmetic progressions, as well an its multidimensional and IP versions. One interesting property of the HalesJewett’s theorem is that … Continue reading
Arithmetic progressions and the affine semigroup
— 1. Introduction — Van der Waerden’s theorem on arithmetic progressions states that, given any finite partition of , one of the cells contains arbitrarily long arithmetic progressions. For the sake of completion we give a precise formulation. Theorem 1 … Continue reading
Posted in Combinatorics, Topological Dynamics
Tagged proximal system, Szemeredi's theorem, van der waerden
1 Comment
Brauer’s theorem and a coloring trick of Bergelson
— 1. Introduction — Ramsey theory concerns essentially two types of results: coloring and density results. Coloring results state that given a finite partition of some structured set (usually ) one of the cells in the partition still has some … Continue reading
Posted in Combinatorics, Ergodic Theory, Ramsey Theory, Tool
Tagged Bergelson, coloring, coloring trick, density, ramsey theory, Szemeredi's theorem, van der waerden
3 Comments
Double van der Waerden
— 1. Introduction — In a previous post I presented a proof of van der Waerden’s theorem on arithmetic progressions: Theorem 1 (van der Waerden, 1927) Consider a partition of the set of the natural numbers into finitely many pieces … Continue reading
Posted in Combinatorics, Ramsey Theory
Tagged arithmetic progressions, geometric progressions, Ultrafilters, van der waerden
4 Comments
HalesJewett Theorem
— 1. Introduction — One of the earliest posts I wrote on this blog contained a proof of the van der Waerden’s Theorem on arithmetic progressions. That proof was topological in nature and illustrated the interesting relation between some problems … Continue reading
Posted in Combinatorics, Ramsey Theory
Tagged arithmetic progressions, ramsey theory, van der waerden
3 Comments