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Tag Archives: ramsey theory
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
Ramsey, Hindman and MillikenTaylor Theorems
Ramsey’s theorem is probably the most famous result of Ramsey theory, giving it its name. Essentially it states that a finite coloring of a certain structure contains a monochromatic copy of the original structure, in this case the structure being … Continue reading
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
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
6 Comments
Properties of ultrafilters and a Theorem on arithmetic combinatorics
A Theorem of Schur (one of the earliest results in Ramsey Theory) asserts that given any finite coloring of the set of natural numbers , there exist of the same color such that also has the same color. As a … Continue reading
Posted in Combinatorics, Ramsey Theory
Tagged Hindman, ramsey theory, rings, StoneCech compactification, Ultrafilters
13 Comments
Convergence along ultrafilters – part II
This is the second of a series of two post whose aim is to prove the following recurrence theorem. Recall that a measure preserving system (shortened to m.p.s.) is a quadruple , where is a probability space and preserves the … Continue reading
Posted in Combinatorics, Ergodic Theory, Ramsey Theory
Tagged plim, ramsey theory, recurrence, Ultrafilters, van der Corput
1 Comment
Convergence along ultrafilters
— 1. Introduction — On my previous post about recurrent theorems I stated Khintchine’s theorem and Sarkozy’s theorem. There I classified Khintchine’s theorem as a theorem about large intersections and Sarkozy’s theorem as a theorem about large recurrent times. This … Continue reading
Posted in Ergodic Theory, Ramsey Theory, Tool
Tagged plim, ramsey theory, recurrence, Ultrafilters, van der Corput
8 Comments