Dona Strauss

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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 this(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)
Let S be a semigroup, and let ` (S) be the Banach algebra which is the semigroup algebra of S. We shall study the structure of this Banach algebra and of its second dual. We shall determine exactly when ` (S) is amenable as a Banach algebra, and shall discuss its amenability constant, showing that there are ‘forbidden values’ for this constant. The second(More)
We investigate when the set of finite products of distinct terms of a sequence 〈xn〉n=1 in a semigroup (S, ·) is large in any of several standard notions of largeness. These include piecewise syndetic, central , syndetic, central* , and IP*. In the case of a “nice” sequence in (S, ·) = (N,+) one has that FS(〈xn〉n=1) has any or all of the first three(More)
Recently, in conversation with Erdős, Hajnal asked whether or not for any triangle-free graph G on the vertex set N, there always exists a sequence 〈xn〉n=1 so that whenever F and H are distinct finite nonempty subsets of N, {Σn∈F xn,Σn∈H xn} is not an edge of G (that is, FS(〈xn〉n=1) is an independent set). We answer this question in the negative. We also(More)
A u × v matrix A is image partition regular provided that, whenever N is finitely colored, there is some ~x ∈ N 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. ∗This author acknowledges support(More)
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(More)