Van Der Waerden's Theorem Research Articles

Polynomial extension of some symmetric partition regular structures

The study of symmetric structures is a new trend in combinatorial number theory. Recently in [9], Di Nasso proved symmetrized versions of some classical theorems like Hindman's theorem, Van der Waerden's theorem and Deuber's theorem. This opens the question of which other classical theorems can be symmetrized, as well as if other symmetric operations allow for such generalizations. Here we give positive answers to both questions, by showing that symmetrization of the polynomial extension of Van der Waerden's and Deuber's Theorem is possible.

Central Sets Theorem in arbitrary semigroup

In this article we provide a version of the Central Sets Theorem, which was originally introduced by H. Furstenberg and considered to be a joint extension of Finite Sum Theorem and van der Waerden's theorem. The original version of this theorem uses nitely many sequences of functions, but here we will provide a version which uses matrices instead of taking sequences.

Fermat's Last Theorem, Schur's Theorem (in Ramsey Theory), and the infinitude of the primes

Alpoge and Granville (separately) gave novel proofs that the primes are infinite that use Ramsey Theory. In particular, they use Van der Waerden's Theorem and some number theory. We prove the primes are infinite using an easier theorem from Ramsey Theory, namely Schur's Theorem, and some number theory (Elsholtz independently obtained the same proof that the primes were infinite). In particular, we use the n=3 case of Fermat's last theorem. We also apply our method to show other domains have an infinite number of irreducibles.

The Green-Tao Theorem and the Infinitude of Primes in Domains

We first prove an elementary analogue of the Green-Tao Theorem. The celebrated Green-Tao Theorem states that there are arbitrarily long arithmetic progressions in the set of prime numbers. In fact, we show the Green-Tao Theorem for polynomial rings over integral domains with several variables. Using the Generalized Polynomial van der Waerden Theorem, we also prove that in an infinite unique factorization domain, if the cardinality of the set of units is strictly less than that of the domain, then there are infinitely many prime elements. Moreover, we deduce the infinitude of prime numbers in the positive integers using polynomial progressions of length three. In addition, using unit equations, we provide two more proofs of the infinitude of prime numbers. Finally, we give a new proof of the divergence of the sum of reciprocals of all prime numbers.

Discordant sets and ergodic Ramsey theory

We explore the properties of non-piecewise syndetic sets with positive upper density, which we call "discordant", in countably infinite amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerden's theorem and Szemer\'{e}di's theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.

Random Van der Waerden Theorem

In this paper, we prove a sparse random analogue of the Van der Waerden Theorem. We show that, for all $r > 2$ and all $q_1 \geq q_2 \geq \dotsb \geq q_r \geq 3 \in \mathbb{N}$, $n^{-\frac{q_2}{q_1(q_2-1)}}$ is a threshold for the following property: For every $r$-coloring of the $p$-random subset of $\{1,\dotsc,n\}$, there exists a monochromatic $q_i$-term arithmetic progression colored $i$, for some $i$. This extends the results of Rödl and Ruciński for the symmetric case $q_1 = q_2 = \dotsb = q_r$. The proof of the 1-statement is based on the Hypergraph Container Method by Balogh, Morris and Samotij and Saxton and Thomason. The proof of the 0-statement is an extension of Rödl and Ruciński's argument for the symmetric case.

Minimum Number of Colours to Avoid k-Term Monochromatic Arithmetic Progressions

By recalling van der Waerden theorem, there exists a least a positive integer w=w(k;r) such that for any n≥w, every r-colouring of [1,n] admits a monochromatic k-term arithmetic progression. Let k≥2 and rk(n) denote the minimum number of colour required so that there exists a rk(n)-colouring of [1,n] that avoids any monochromatic k-term arithmetic progression. In this paper, we give necessary and sufficient conditions for rk(n+1)=rk(n). We also show that rk(n)=2 for all k≤n≤2(k−1)2 and give an upper bound for rp(pm) for any prime p≥3 and integer m≥2.

A variant of the proof of Van der Waerden's theorem by Furstenberg

Let R R be a commutative ring with identity. In this paper, for a given monotone decreasing positive sequence and an increasing sequence of subsets of R R , we will define a metric on R R using them. Then, we will use this kind of metric to obtain a variant of the proof of Van der Waerden's theorem by Furstenberg [3].

A short proof of the canonical polynomial van der Waerden theorem

Nous présentons une nouvelle démonstration courte du théorème de van der Waerden polynomial canonique, récemment établi par Girão.

Szemerédi’s proof of Szemerédi’s theorem

In 1975, Szemeredi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der Waerden theorem and a lemma now known as the Szemeredi regularity lemma, together with a delicate analysis (based ultimately on double counting arguments) of limiting densities of sets along multidimensional arithmetic progressions. In this note we present an arrangement of this proof that incorporates a number of notational and technical simplifications. Firstly, we replace the use of the regularity lemma by that of the simpler ``weak regularity lemma'' of Frieze and Kannan. Secondly, we extract the key inductive steps at the core of Szemeredi's proof (referred to as ``Lemma 5'', ``Lemma 6'', and ``Fact 12'' in that paper) as stand-alone theorems that can be stated with less notational setup than in the original proof, in particular involving only (families of) one-dimensional arithmetic progressions, as opposed to multidimensional arithmetic progressions. Thirdly, we abstract the analysis of limiting densities along the (now one-dimensional) arithmetic progressions by introducing the notion of a family of arithmetic progressions with the ``double counting property''. We also present a simplified version of the argument that is capable of establishing Roth's theorem on arithmetic progressions of length three.

On the Ramsey number of the Brauer configuration

We obtain a double exponential bound in Brauer's generalisation of van der Waerden's theorem, which concerns progressions with the same colour as their common difference. Such a result has been obtained independently and in much greater generality by Sanders. Using Gowers' local inverse theorem, our bound is quintuple exponential in the length of the progression. We refine this bound in the colour aspect for three-term progressions, and combine our arguments with an insight of Lefmann to obtain analogous bounds for the Ramsey numbers of certain non-linear quadratic equations.

Open Problems Column

There have been two polls asking theorists (and others) what they thought of P vs NP (and other questions) [4, 5]. Both were written by William Gasarch and appeared in the Complexity Column of SIGACT News, edited by Lane A. Hemaspaandra. They were in 2002 and 2012. Since William Gasarch is fond of Van der Waerden's theorem, you would think the next poll would be 2022; however, Lane asked if it could be a bit earlier so as to be the 100th issue of his column. So here we are. In the past the questions for the poll appeared in Lane's column. This time they appear here in the Open Problems Column, which makes sense since, well, P vs NP is indeed an open problem.

Squares in Arithmetic Progressions and Infinitely Many Primes

We give a new proof that there are infinitely many primes, relying on van derWaerden's theorem for coloring the integers, and Fermat's theorem that there cannot be four squares in an arithmetic progression. We go on to discuss where else these ideas have come together in the past.

Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

AbstractFor a given graph F we consider the family of (finite) graphs G with the Ramsey property for F, that is the set of such graphs G with the property that every two‐coloring of the edges of G yields a monochromatic copy of F. For F being a triangle Friedgut, Rödl, Ruciński, and Tetali (2004) established the sharp threshold for the Ramsey property in random graphs. We present a simpler proof of this result which extends to a more general class of graphs F including all cycles.The proof is based on Friedgut's criteria (1999) for sharp thresholds and on the recently developed container method for independent sets in hypergraphs by Saxton and Thomason and by Balogh, Morris and Samotij. The proof builds on some recent work of Friedgut et al. who established a similar result for van der Waerden's theorem.

Revisiting the nilpotent polynomial Hales–Jewett theorem

Answering a question posed by Bergelson and Leibman in [6], we establish a nilpotent version of the Polynomial Hales–Jewett Theorem that contains the main theorem in [6] as a special case. Important to the formulation and the proof of our main theorem is the notion of a relative syndetic set (relative with respect to a closed non-empty subsets of βG) [25]. As a corollary of our main theorem we prove an extension of the restricted van der Waerden Theorem to nilpotent groups, which involves nilprogressions.

Multi-recurrence and van der Waerden systems

We explore recurrence properties arising from dynamical approach to combinatorial problems like the van der Waerden Theorem. We describe relations between these properties, study their consequences for dynamics, and demonstrate connections to combinatorial problems. In particular, we present a measure-theoretical analog of a result of Glasner on multi-transitivity in topological weakly mixing minimal maps. We also obtain a dynamical proof of the existence of a C-set with zero Banach density.

An analogue of a theorem of van der Waerden, and its application to two-distance preserving mappings

The van der Waerden's theorem reads that an equilateral pentagon in Euclidean 3-space $\Bbb E^3$ with all diagonals of the same length is necessarily planar and its vertex set coincides with the vertex set of some convex regular pentagon. We prove the following many-dimensional analogue of this theorem: for $n\geqslant 2$, every $n$-dimensional cross-polytope in $\Bbb E^{2n-2}$ with all diagonals of the same length and all edges of the same length necessarily lies in $\Bbb E^n$ and hence is a convex regular cross-polytope. We also apply our theorem to the study of two-distance preserving mappings of Euclidean spaces.

A sharp threshold for van der Waerden's theorem in random subsets

We establish sharpness for the threshold of van der Waerden's theorem in random subsets of $\mathbb{Z}/n\mathbb{Z}$. More precisely, for $k\geq 3$ and $Z\subseteq \mathbb{Z}/n\mathbb{Z}$ we say $Z$ has the van der Waerden property if any two-colouring of $Z$ yields a monochromatic arithmetic progression of length $k$. R\"odl and Ruci\'nski (1995) determined the threshold for this property for any k and we show that this threshold is sharp. The proof is based on Friedgut's criteria (1999) for sharp thresholds, and on the recently developed container method for independent sets in hypergraphs by Balogh, Morris and Samotij (2015) and by Saxton and Thomason (2015).

Van der Waerden and the Primes

In this note we prove the infinitude of the primes via an application of van der Waerden's theorem.

Convergence in van der Waerden and Hindman spaces

We consider four classes of topological spaces which are defined with the aid of convergence with respect to ideals on N. All these classes are subclasses of countably compact spaces, and two of them are also subclasses of sequentially compact spaces. In the first part of the paper (Sections 1 and 2) we prove some properties of these classes. In the second part of the paper (Sections 3 and 4) we focus on spaces defined by two particular ideals connected with well known theorems in combinatorics, namely van der Waerden's theorem and Hindman's theorem. The main aim of this part of the paper is to show that two classes of the considered spaces coincide for the van der Waerden ideal and Hindman ideal respectively.