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We show that if (S, +) is a commutative semigroup which can be embedded in the circle group T, in particular if S = (N, +), then all nonprincipal, strongly summable ultrafilters on S are sparse and can be written as sums in βS only trivially. We develop a simple condition on a strongly summable ultrafilter which guarantees that it is sparse and show that(More)
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. While in the finite case partition(More)
Recently, in conversation with Erd} os, Hajnal asked whether or not for any triangle-free graph G on the vertex set N , there always exists a sequence hx n i 1 n=1 so that whenever F and H are distinct nite nonempty subsets of N , f n2F x n ; n2H x n g is not an edge of G (that is, FS(hx n i 1 n=1) is an independent set). We answer this question in the(More)
As published there were gaps in the proofs that (a) implies (d) and that (b) implies (c) in Theorem 2.10. These gaps have been filled here. Other than that, to the best of my knowledge, this is the final version as it was submitted to the publisher. – NH Abstract A u × v matrix A is image partition regular provided that, whenever N is finitely colored,(More)
Central sets in semigroups are known to have very rich combinato-rial structure, described by the " Central Sets Theorem ". It has been unknown whether the Central Sets Theorem in fact characterizes central sets, and if not whether some other combinatorial characterization could be found. We derive here a combinatorial characterization of central sets and(More)
We consider the question of the existence of a nontrivial continuous homomorphism from (N +) into N = NnN. This problem is known to be equivalent to the existence of distinct p and q in N satisfying the equations p + p = q = q + q = q + p = p + q. We obtain certain restrictions on possible values of p and q in these equations and show that the existence of(More)
Previous research extending over a few decades has established that multi-plicatively large sets (in any of several interpretations) must have substantial additive structure. We investigate here the question of how much multiplicative structure can be found in additively large sets. For example, we show that any translate of a set of finite sums from an(More)
Deuber's Theorem says that, given any m, p, c, r in N, there exist n, q, µ in N such that whenever an (n, q, c µ)-set is r-coloured, there is a monochrome (m, p, c)-set. This theorem has been used in conjunction with the algebraic structure of the Stone-ˇ Cech compactification βN of N to derive several strengthenings of itself. We present here an algebraic(More)
A finite or infinite matrix A with rational entries (and only finitely many non-zero entries in each row) is called image partition regular if, whenever the natural numbers are finitely coloured, there is a vector x, with entries in the natural numbers, such that Ax is monochromatic. Many of the classical results of Ramsey theory are naturally stated in(More)