
Recent Posts
Tag Archives: arithmetic progressions
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
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
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
Covering the natural numbers with generalized arithmetic progressions
In this short post I will present some curious facts I came across recently. Given a real number construct the set , where, as usual, denotes the floor function. Note that if is an integer, then is an (infinite) arithmetic … Continue reading