Dona Strauss

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

Some new results in multiplicative and additive Ramsey theory

x

Central Sets and Their Combinatorial Characterization

with

Image partition regular matrices - bounded solutions and preservation of largeness

Ax=0

Independent Finite Sums forKm-Free Graphs

. 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.

Cancellation in the Stone–Čech 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.

Partition regularity without the columns property

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.

Banach Spaces of Continuous Functions as Dual Spaces

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sucien tly well behaved semigroups including (N; +) and (N; ). It is known that sets with positive multiplicative density must contain arbitrarily large geoarithmetic progressions, that is, sets of the form fr j (a + id) : i; j 2 f0; 1; : : : ; kgg. We establish some combined additive and multiplicative Ramsey Theoretic consequences of known algebraic results in the semigroups ( N; +) and ( N; ), derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that in any nite partition of N there must be, for each k, sets of the form fb(a+id) j : i; j 2 f0; 1; : : : ; kgg together with d, the arithmetic progression fa + id : i 2 f0; 1; : : : ; kgg, and the geometric progression fbd j : j 2 f0; 1; : : : ; kgg in one cell of the partition.