Related papers: Enumeration of three term arithmetic progressions …

A famous theorem of Szemer\'edi asserts that any set of integers of positive upper density will contain arbitrarily long arithmetic progressions. In its full generality, we know of four types of arguments that can prove this theorem: the…

A famous theorem of Szemer\'edi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general…

Szemer\'edi's Theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain…

A famous theorem of Szemer\'edi asserts that all subsets of the integers with positive upper density will contain arbitrarily long arithmetic progressions. There are many different proofs of this deep theorem, but they are all based on a…

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of…

This is an exposition of the combinatorial proof of the density Hales--Jewett theorem, due to D.\,H.\,J. Polymath in 2012. The theorem says that for given $\de>0$ and $k$, for every $n>n_0$ every set $A\sus\{1,2,\ds,k\}^n$ with $|A|\ge\de…

Green and Tao famously proved in 2005 that any subset of the primes of fixed positive density contains arbitrarily long arithmetic progressions. Green had previously shown that in fact any subset of the primes of relative density tending to…

Furstenberg-Weiss have extended Szemer\'edi's theorem on arithmetic progressions to trees by showing that a large subset of the tree contains arbitrarily long arithmetic subtrees. We study higher dimensional versions that analogously extend…

Szemer\'edi's regularity lemma is a basic tool in graph theory, and also plays an important role in additive combinatorics, most notably in proving Szemer\'edi's theorem on arithmetic progressions . In this note we revisit this lemma from…

The celebrated Green-Tao theorem states that there are arbitrarily long arithmetic progressions in the primes. One of the main ingredients in their proof is a relative Szemer\'edi theorem which says that any subset of a pseudorandom set of…

One of the "deepest" theorems in mathematics is Endre Szemer\'edi's theorem about the inevitability of arithmetical progressions. Here we try to nibble at it, by doing "finite" analogs. This is already interesting for its own sake, but we…

In 1975 Szemer\'edi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a…

Van der Waerden's theorem asserts that if you color the natural numbers with, say, five different colors, then you can always find arbitrarily long sequences of numbers that have the same color and that form an arithmetic progression.…

The study of many problems in additive combinatorics, such as Szemer\'edi's theorem on arithmetic progressions, is made easier by first studying models for the problem in F_p^n for some fixed small prime p. We give a number of examples of…

Shortly after Szemer\'edi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, Furstenberg gave a new proof of this theorem using ergodic theory. This gave rise to the field of ergodic Ramsey…

Green proved an arithmetic analogue of Szemer\'edi's celebrated regularity lemma and used it to verify a conjecture of Bergelson, Host, and Kra which sharpens Roth's theorem on three-term arithmetic progressions in dense sets. It shows that…

Recently Conlon, Fox, and the author gave a new proof of a relative Szemer\'edi theorem, which was the main novel ingredient in the proof of the celebrated Green-Tao theorem that the primes contain arbitrarily long arithmetic progressions.…

In 1975 Szemer\'edi proved the long-standing conjecture of Erd\H{o}s and Tur\'an that any subset of $\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemer\'edi's proof was entirely…

We consider, over both the integers and finite fields, Szemer\'{e}di's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of…

Using the density-increment strategy of Roth and Gowers, we derive Szemeredi's theorem on arithmetic progressions from the inverse conjectures GI(s) for the Gowers norms, recently established by the authors and Ziegler.