On Milliken-Taylor ultrafilters

Abstract. We show that there may be a Milliken-Taylor ultrafilter with infinitely many near coherence classes of ultrafilters in its projection to ω, answering a question by López-Abad. We show that k-coloured Milliken-Taylor ultrafilters have at least k + 1 near coherence classes of ultrafilters in its projection to ω. We show that the Mathias forcing with a Milliken-Taylor ultrafilter destroys all Milliken-Taylor ultrafilters from the ground model. Milliken-Taylor ultrafilters and their projections We answer a question of López-Abad whether there can be more than two near coherence classes of ultrafilters in the core of a Milliken-Taylor ultrafilter. Then we show that in Milliken Taylor ultrafilter with k colours there are k + 1 near coherence classes in its projection to ω, generalising a result of Blass In the rest of this introductory section we review part of the relevant background. Our nomenclature follows [10] an

Centrally Image partition Regularity near 0

The notion of Image partition regularity near zero was first introduced by De and Hindman. It was shown there that like image partition regularity over $\mathbb{N}$ the main source of infinite image partition regular matrices near zero are Milliken- Taylor matrices. But Milliken- Taylor matrices are far apart to have images in central sets. In this regard the notion of centrally image partition regularity was introduced. In the present paper we propose the notion centrally partition regular matrices near zero for dense sub semigroup of (\ber^+,+) which are different from centrally partition regular matrices unlike finite cases

Infinite monochromatic patterns in the integers

We show the existence of several infinite monochromatic patterns in the integers obtained as values of suitable symmetric polynomials; in particular, we obtain extensions of both the additive and multiplicative versions of Hindman's theorem. These configurations are obtained by means of suitable symmetric polynomials that mix the two operations. The simplest example is the following. For every finite coloring N=C1∪…∪Cr there exists an infinite increasing sequence a<… such that all elements below are monochromatic: a,b,c,…,a+b+ab,a+c+ac,b+c+bc,…,a+b+c+ab+ac+bc+abc,…. The proofs use tools from algebra in the space of ultrafilters βZ