Category Archives: Ramsey Theory

Hales-Jewett and a generalized van der Warden Theorems

The Hales-Jewett 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 Hales-Jewett’s theorem is that … Continue reading

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Ramsey, Hindman and Milliken-Taylor 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

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Pomerance Theorem on colinear points in certain paths in a two dimensional lattice

— 1. Introduction — Van der Waerden’s theorem (to which I gave two proofs in previous posts on this blog) states that if one colors the positive integers with finitely many colors, then one can always find a monochromatic arithmetic … Continue reading

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Szemerédi’s Theorem Part III – Precise definitions

This is the third in a series of six posts on Szemerédi’s theorem. In the previous post I outlined the ideas of the ergodic theoretical proof by Furstenberg. In this post I will set up the machinery and give the … Continue reading

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Szemerédi’s Theorem Part I – Equivalent formulations

The theorem of van der Waerden on arithmetic progressions, whose precise statement and proof can be found in a previous post of mine, states that in a finite partition of the set of positive integers, one of the pieces contains … Continue reading

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On {x+y,xy} patterns in large sets of countable fields

Vitaly Bergelson and I have recently uploaded to the arXiv our joint paper `On patterns in large sets of countable fields‘. We prove a result concerning certain monochromatic structures in countable fields and a corresponding density version. Schur’s Theorem, proved … Continue reading

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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

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