Partition Regular Research Articles

A Transference Approach to a Roth-Type Theorem in the Squares

We show that any subset of the squares of positive relative upper density contains nontrivial solutions to a translation-invariant linear equation in five or more variables, with explicit quantitative bounds. As a consequence, we establish the partition regularity of any diagonal quadric in five or more variables whose coefficients sum to zero. Unlike previous approaches, which are limited to equations in seven or more variables, we employ transference technology of Green to import bounds from the linear setting.

On some variations of coloring problems of infinite words

Given a finite coloring (or finite partition) of the free semigroup A+ over a set A, we consider various types of monochromatic factorizations of right sided infinite words x∈Aω. Some stronger versions of the usual notion of monochromatic factorization are introduced. A factorization is called sequentially monochromatic when concatenations of consecutive blocks are monochromatic. A sequentially monochromatic factorization is called ultra monochromatic if any concatenation of arbitrary permuted blocks of the factorization has the same color of the single blocks. We establish links, and in some cases equivalences, between the existence of these factorizations and fundamental results in Ramsey theory including the infinite Ramsey theorem, Hindman's finite sums theorem, partition regularity of IP sets and the Milliken–Taylor theorem. We prove that for each finite set A and each finite coloring φ:A+→C, for almost all words x∈Aω, there exists y in the subshift generated by x admitting a φ-ultra monochromatic factorization, where “almost all” refers to the Bernoulli measure on Aω.

Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem

By using nonstandard analysis, and in particular iterated hyper-extensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for applications in Ramsey theory of numbers. To illustrate the use of our technique, we give a (rather) short proof of Milliken-Taylor's Theorem, and a ultrafilter version of Rado's theorem about partition regularity of diophantine equations.

Partition regularity in the rationals

A system of homogeneous linear equations with integer coefficients is partition regular if, whenever the natural numbers are finitely coloured, the system has a monochromatic solution. The Finite Sums theorem provided the first example of an infinite partition regular system of equations. Since then, other such systems of equations have been found, but each can be viewed as a modification of the Finite Sums theorem.We present here a new infinite partition regular system of equations that appears to arise in a genuinely different way. This is the first example of a partition regular system in which a variable occurs with unbounded coefficients. A modification of the system provides an example of a system that is partition regular over Q but not N, settling another open problem.

Partition regularity with congruence conditions

An infinite integer matrix A is called image partition regular if, whenever the natural numbers are finitely coloured, there is an integer vector x such that Ax is monochromatic. Given an image partition regular matrix A, can we also insist that each variable x_i is a multiple of some given d_i? This is a question of Hindman, Leader and Strauss. Our aim in this short note is to show that the answer is negative. As an application, we disprove a conjectured equivalence between the two main forms of partition regularity, namely image partition regularity and kernel partition regularity.

Partition regularity and the primes

The purpose of this Note is to point out, as a simple yet nice consequence of Green and Taoʼs program on counting linear patterns in the primes and Deuberʼs work on partition regularity, that if a system of equations is partition regular over the positive integers, then it is also partition regular over the sets {p−1:p prime} as well as {p+1:p prime}. This answers a question of Li and Pan.

A topological generalization of partition regularity

In 1939, Richard Rado showed that any complex matrix is partition regular over C if and only if it satisfies the columns condition. Recently, Hogben and McLeod explored the linear algebraic properties of matrices satisfying partition regularity. We further the discourse by generalizing the notion of partition regularity beyond systems of linear equations to topological surfaces and graphs. We begin by defining, for an arbitrary matrix 8, the metric space (M8, ). Here, M8 is the set of all matrices equivalent to 8 that are (not) partition regular if 8 is (not) partition regular; and for elementary matrices, Ei and Fj , we let. A; B/D minfmD lCkV BD E1:::El AF1:::Fkg. Subsequently, we illustrate that partition regularity is in fact a local property in the topological sense, and uncover some of the properties of partition regularity from this perspective. We then use these properties to establish that all compact topological surfaces are partition regular.

Equations resolving a conjecture of Rado on partition regularity

A linear equation L is called k-regular if every k-coloring of the positive integers contains a monochromatic solution to L. Richard Rado conjectured that for every positive integer k, there exists a linear equation that is ( k − 1 ) -regular but not k-regular. We prove this conjecture by showing that the equation ∑ i = 1 k − 1 2 i 2 i − 1 x i = ( − 1 + ∑ i = 1 k − 1 2 i 2 i − 1 ) x 0 has this property. This conjecture is part of problem E14 in Richard K. Guy's book “Unsolved Problems in Number Theory”, where it is attributed to Rado's 1933 thesis, “Studien zur Kombinatorik”.

Universally Image Partition Regularity

Many of the classical results of Ramsey Theory, for example Schur's Theorem, van der Waerden's Theorem, Finite Sums Theorem, are naturally stated in terms of image partition regularity of matrices. Many characterizations are known of image partition regularity over ${\Bbb N}$ and other subsemigroups of $({\Bbb R},+)$. In this paper we introduce a new notion which we call universally image partition regular matrices, which are in fact image partition regular over all semigroups and everywhere. We also prove that such matrices exist in abundance.

Image partition regularity near zero

Many of the classical results of Ramsey Theory are naturally stated in terms of image partition regularity of matrices. Many characterizations are known of image partition regularity over N and other subsemigroups of ( R , + ) . We study several notions of image partition regularity near zero for both finite and infinite matrices, and establish relationships which must hold among these notions.

Image partition regularity of affine transformations

If u , v ∈ N , A is a u × v matrix with entries from Q , and b → ∈ Q u , then ( A , b → ) determines an affine transformation from Q v to Q u by x → ↦ A x → + b → . In 1933 and 1943 Richard Rado determined precisely when such transformations are kernel partition regular over N , Z , or Q , meaning that whenever the nonzero elements of the relevant set are partitioned into finitely many cells, there is some element of the kernel of the transformation with all of its entries in the same cell. In 1993 the first author and Imre Leader determined when such transformations with b → = 0 ¯ are image partition regular over N , meaning that whenever N is partitioned into finitely many cells, there is some element of the image of the transformation with all of its entries in the same cell. In this paper we characterize the image partition regularity of such transformations over N , Z , or Q for arbitrary b → .

Independence for partition regular equations

A matrix A is said to be partition regular ( PR) over a subset S of the positive integers if whenever S is finitely coloured, there exists a vector x, with all elements in the same colour class in S, which satisfies A x = 0 . We also say that S is PR for A. Many of the classical theorems of Ramsey Theory, such as van der Waerden's theorem and Schur's theorem, may naturally be interpreted as statements about partition regularity. Those matrices which are partition regular over the positive integers were completely characterised by Rado in 1933. Given matrices A and B, we say that A Rado-dominates B if any set which is PR for A is also PR for B. One trivial way for this to happen is if every solution to A x = 0 actually contains a solution to B y = 0 . Bergelson, Hindman and Leader conjectured that this is the only way in which one matrix can Rado-dominate another. In this paper, we prove this conjecture for the first interesting case, namely for 1 × 3 matrices. We also show that, surprisingly, the conjecture is not true in general.

SPARSE PARTITION REGULARITY

Our aim in this paper is to prove Deuber's conjecture on sparse partition regularity, that for every $m$, $p$ and $c$ there exists a subset of the natural numbers whose $(m,p,c)$-sets have high girth and chromatic number. More precisely, we show that for any $m$, $p$, $c$, $k$ and $g$ there is a subset $S$ of the natural numbers that is sufficiently rich in $(m,p,c)$-sets that whenever $S$ is $k$-coloured there is a monochromatic $(m,p,c)$-set, yet is so sparse that its $(m,p,c)$-sets do not form any cycles of length less than $g$.Our main tools are some extensions of Nešetřil–Rödl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.

FORBIDDEN DISTANCES IN THE RATIONALS AND THE REALS

Our main aim in this paper is to show that there is a partition of the reals into finitely many classes with ‘many’ forbidden distances, in the following sense: for each positive real x , there is a natural number n such that no two points in the same class are at distance $x/n$ . In fact, more generally, given any infinite set $\{ c_n:n of positive rationals, there is a partition of the reals into three classes such that for each positive real x , there is some n such that no two points in the same class are at distance $c_nx$ . This result is motivated by some questions in partition regularity.

Assembly of sodium dodecyl sulfate at the wool/water interface

The adsorption isotherms for sodium dodecyl sulfate (SDS) adsorption from aqueous solutions (pH 4.69) onto wool/water interfaces at 15, 25 and 35 °C, respectively, are satisfactorily expressed in linear plots based on a stoichiometric displacement model of adsorption (SDM-A). The studies on adsorption parameters of SDM-A suggest that assembly of SDS molecules on the wool/solution interface follows the partition regularity and stoichiometric displacement relation between the two phases based on the SDM-A. The fact that the general displacement adsorption heats obtained in the calorimetric measurement and the enthalpies calculated by the thermodynamics of SDM-A in the presentation are well in accordance with each other shows that thermodynamic function fraction study of SDM-A is reliable. The studies on thermodynamic parameters and function fractions suggest that at the start of adsorption, the SO 4 − groups occupy the sites from which the water molecules are released and simultaneously driven to the edges of alkyl chains closed to polar heads. The surface hydrophobic interactions increase in the way that the adsorbed SDS molecules on the surface favor to form bilayer micelles while every individual SDS molecule in the surface micelles may attach to surface first. The investigation of fluorescence probe of pyrene indicates that surface micelle formation begins in lower surface coverage while the aggregation numbers of micelles grow at high surface coverage.

Open Problems in Partition Regularity

A finite or infinite matrix A with rational entries is called partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic vector x with . Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular.While in the finite case partition regularity is well understood, very little is known in the infinite case. Our aim in this paper is to present some of the natural and appealing open problems in the area.

Independent Deuber sets in graphs on the natural numbers

We show that for any k, m, p, c, if G is a K k -free graph on N then there is an independent set of vertices in G that contains an ( m, p, c)-set. Hence if G is a K k -free graph on N , then one can solve any partition regular system of equations in an independent set. This is a common generalization of partition regularity theorems of Rado (who characterized systems of linear equations A x = 0 a solution of which can be found monochromatic under any finite coloring of N ) and Deuber (who provided another characterization in terms of ( m, p, c)-sets and a partition theorem for them), and of Ramsey's theorem itself.

Infinite partition regular matrices: solutions in central sets

A finite or infinite matrix A is image partition regular provided that whenever N is finitely colored, there must be some £ with entries from N such that all entries of AS are in the same color class. In contrast to the finite case, infinite image partition regular matrices seem very hard to analyze: they do not enjoy the closure and consistency properties of the finite case, and it is difficult to construct new ones from old. In this paper we introduce the stronger notion of central image partition regularity, meaning that A must have images in every central subset of N. We describe some classes of centrally image partition regular matrices and investigate the extent to which they are better behaved than ordinary image partition regular matrices. It turns out that the centrally image partition regular matrices are closed under some natural operations, and this allows us to give new examples of image partition regular matrices. In particular, we are able to solve a vexing open problem by showing that whenever N is finitely colored, there must exist injective sequences ∞ n =0 and ∞ n =0 in N with all sums of the forms x n + x m and z n + 2z m with n < m in the same color class. This is the first example of an image partition regular system whose regularity is not guaranteed by the Milliken-Taylor Theorem, or variants thereof.

Image partition regular matrices—bounded solutions and preservation of largeness

A u× v matrix A is image partition regular provided that, whenever N is finitely colored, there is some x → ∈ N v with all entries of A x → monochrome. Image partition regular matrices are a natural way of representing some of the classic theorems of Ramsey Theory, including theorems of Hilbert, Schur, and van der Waerden. We present here some new characterizations and consequences of image partition regularity and investigate some issues raised by these. One of our characterizations is that the image partition regular matrices are precisely those that preserve a certain notion of largeness (“central sets”)—we examine what happens for other well known notions of largeness. Another property of image partition regular matrices is that (except in trivial cases) the entries of A x → may be chosen to be distinct—we investigate when we may choose the entries to be “close together” or “far apart” in various senses.

Rado's theorem for commutative rings

We establish that an analogue of Rado's “columns condition” is sufficient for the partition regularity of a homogeneous system of equations Ax = 0 in any commutative ring R. (A system of equations with coefficients in R is partition regular provided that for any finite partition of Rβ{0}, one cell contains a solution set for that system.) We show that for a wide class of commutative rings, the condition is also necessary. We investigate briefly solutions to a nonhomogeneous system Ax = b. We also show that in any infinite integral domain one can obtain, in one cell of any finite partition, solution sets for any sequence of systems of equations satisfying the columns condition and all finite sums choosing at most one term from each solution set.