Neil Hindman

Finite sums from sequences within cells of a partition of N

Abstract The principal result of this paper establishes the validity of a conjecture by Graham and Rothschild. This states that, if the natural numbers are divided into two classes, then there is a sequence drawn from one of those classes such that all finite sums of distinct members of that sequence remain in the same class.

Partitions and sums and products of integers

The principal result of the paper is that, if r < u and (A¡)i<r is a partition of u, then there exist i < r and infinite subsets B and C of to such that 2 F e A¡ and IIG G A¡ whenever F and G are finite nonempty subsets of B and C respectively. Conditions on the partition are obtained which are sufficient to guarantee that B and C can be chosen equal in the above statement, and some related finite questions are investigated.

Open Problems in Partition Regularity

A finite or infinite matrix

Multiplicative structures in additively large sets

A

Infinite partition regular matrices: solutions in central sets

with rational entries is called partition regular if, whenever the natural numbers are finitely coloured, there is a monochromatic vector

Partition Regular Structures Contained in Large Sets Are Abundant

x

On strongly summable ultrafilters and union ultrafilters

with

Infinite partition regular matrices

Ax=0

On IP* sets and central sets

. 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.

Filters and the weak almost periodic compactification of a discrete semigroup

Previous research extending over a few decades has established that multiplicatively 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 infinite sequence must contain all of the initial products from another infinite sequence. And, as a corollary of a result of Renling Jin, we show that if A and B have positive upper Banach density, then A + B contains all of the initial products from an infinite sequence. We also show that if a set has a complement which is not additively piecewise syndetic, then any translate of that set is both additively and multiplicatively large in several senses.We investigate whether a subset of N with bounded gaps--a syndetic set--must contain arbitrarily long geometric progressions. We believe that we establish that this is a significant open question.