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Our aim in this paper is to show that, for any k, there is a finite colouring of the set of rationals whose denominators contain only the first fc primes such that no infinite set has all of its finite sums and products monochromatic. We actually prove a 'uniform' form of this: there is a finite colouring of the rationals with the property that no infinite set whose denominators contain only finitely many primes has all of its finite sums and products monochromatic. We also give various other results, including a new short proof of the old result that there is a finite colouring of the naturals such that no infinite set has all of its pairwise sums and products monochromatic. [ABSTRACT FROM AUTHOR]

A factorisation x=u1u2⋯$x = u_1 u_2 \cdots$ of an infinite word x$x$ on alphabet X$X$ is called 'monochromatic', for a given colouring of the finite words X∗$X^*$ on alphabet X$X$, if each ui$u_i$ is the same colour. Wojcik and Zamboni proved that the word x$x$ is periodic if and only if for every finite colouring of X∗$X^*$ there is a monochromatic factorisation of x$x$. On the other hand, it follows from Ramsey's theorem that, for any word x$x$, for every finite colouring of X∗$X^*$ there is a suffix of x$x$ having a monochromatic factorisation. A factorisation x=u1u2⋯$x = u_1 u_2 \cdots$ is called 'super‐monochromatic' if each word uk1uk2⋯ukn$u_{k_1} u_{k_2} \cdots u_{k_n}$, where k1<⋯

We show that there is a rational vector space V such that, whenever V is finitely coloured, there is an infinite set X whose sumset X + X is monochromatic. Our example is the rational vector space of dimension sup{..., ...). This complements a result of Hindman, Leader and Strauss, who showed that the result does not hold for dimension below ...w. So our result is best possible under GCH. [ABSTRACT FROM AUTHOR]

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 Ax = 0. Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular. In the finite case, Rado proved that a matrix is partition regular if and only it satisfies a computable condition known as the columns property. The first requirement of the columns property is that some set of columns sums to zero. In the infinite case, much less is known. There are many examples of matrices with the columns property that are not partition regular, but until now all known examples of partition regular matrices did have the columns property. Our main aim in this paper is to show that, perhaps surprisingly, there are infinite partition regular matrices without the columns property -- in fact, having no set of columns summing to zero. We also make a conjecture that if a partition regular matrix (say with integer coefficients) has bounded row sums then it must have the columns property, and prove a first step towards this. [ABSTRACT FROM AUTHOR]

Abstract: A finite set X in some Euclidean space is called Ramsey if for any k there is a d such that whenever is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory. In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever ‘rival’ conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively—in other words, if all points of the set look the same—our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement. Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest. [Copyright &y& Elsevier]

Abstract: 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 . 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 actually contains a solution to . 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 matrices. We also show that, surprisingly, the conjecture is not true in general. [Copyright &y& Elsevier]

Abstract: It is easy to deduce from Ramsey''s theorem that given positive integers and a finite colouring of the set of positive integers, there exists an injective sequence with all sums of the form () lying in the same colour class. The consistency version of this result, namely that given positive integers and , and a finite colouring of , there exist injective sequences and with all sums of the form and all sums of the form () in the same colour class, was open for some time, being recently proved by Hindman, Leader and Strauss. The proof is long and relies heavily on the structure of the semigroup of ultrafilters on . Our aim in this note is to present a short proof of this result which does not use properties of . Our proof also gives various results not obtainable by the previous method of proof. [Copyright &y& Elsevier]

We show that for any k,m,p,c, if G is a Kk-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 Kk-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 Ax=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. [Copyright &y& Elsevier]

The set 0+ of ultrafilters on (0,1) that converge to 0 is a semigroup under the restriction of the usual operation + on &BetaR[subd] the Stone-Çech compactification of the discrete semigroup (R[subd], +). It is also a subsemigroup of Β((0, 1)[subd]). The interaction of these operations has recently yielded some strong results in Ramsey Theory. Since (0[sup+],.) is an ideal of &Beta((0, 1)[subd],.), much is known about the structure of (0[sup+],.). On the other hand, (0[sup+],+) is far from being an ideal of (ΒR[subd],+) so little about its algebraic structure follows from known results. We characterize here the smallest ideal of (0[sup+],+), its closure, and those sets "central" in (0[sup+],+), that is, those sets which are members of minimal idempotents in (0[sup+],+). We derive new combinatorial applications of those sets that are central in (0[sup+],+). [ABSTRACT FROM AUTHOR]

Abstract: A function f from to is said to be feebly continuous at a point if there exist sequences and with . Dales asked if every function has a point of feeble continuity. Our aim in this short note is to show that (assuming the Continuum Hypothesis) the answer is ‘no’. Dales also asked what happens for functions taking only two values: we show that in this case the answer is ‘yes’. [Copyright &y& Elsevier]

Abstract: 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 but not , settling another open problem. [Copyright &y& Elsevier]