We show by elementary methods that given any finite partition of the set N of positive integers, there is one cell that is both additively and multiplicatively rich. In particular, this cell must cont...

We prove a theorem about idempotents in compact semigroups. This theorem gives a new proof of van der Waerden’s theorem on arithmetic progressions as well as the Hales-Jewett theorem. It also gives an...

We investigate the effect of a variant of Matet forcing on ultrafilters in the ground model and give a characterization of those P –points that survive such forcing, answering a question left open by ...

We discuss methods for identifying random phenomena which are logical consequences of the fact that almost all real numbers are of a high descriptive complexity. These methods enable us to find comple...

We derive a generalization of a theorem of Raimi proving there is a partition of natural numbers with given densities of classes which meet structured translates of any other class of a partition of n...

Utilizing ultrafilters on the setN of natural numbers which have certain special properties, we prove some simultaneous generalizations of Ramsey's Theorem and several single dimension Ramsey-type the...

Ultrafilters are a tool, originating in mathematical logic and general topology, that has steadily found more and more uses in multiple areas of mathematics, such as combinatorics, dynamics, and algeb...

The main result provide a common generalization for Ramsey-type theorems concerning finite colorings of edge sets of complete graphs with vertices in infinite semigroups. We capture the essence of the...

The Erdos sum of reciprocals conjecture is the statement that whenever A is a set of positive integers and ∑∈ A 1/ x = ∞, A contains arbitrarily long arithmetic progressions. It is shown here that thi...

Ramsey theory is best defined by example and the classic example of a Ramsey type theorem is the result of van der Waerden: if the integers Z are partitioned into finitely many sets, one of these cont...

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 qu...

Preliminaries Van der Waerden's theorem Supersets of $AP$ Subsets of $AP$ Other generalizations of $w(k r)$ Arithmetic progressions (mod $m$) Other variations on van der Waerden's theorem Schur's theo...

In their, by now classical, paper ‘Ramsey’s theorem for n-parameter sets’ (Trans. Amer. Math. Soc. 159 (1971), 257–291) Graham and Rothschild introduced a combinatorial structure which turned out be c...

IP* sets and central sets are subsets of ℕ which are known to have rich combinatorial structure. We establish here that structure is significantly richer that was previously known. We also establish t...

Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. Ramsey algebras are structures that satisfy an analogue of Hindman’s theorem. The main ...

Hindman’s theorem says that every finite coloring of the natural numbers has a monochromatic set of finite sums. A Ramsey algebra is a structure that satisfies an analogue of Hindman’s theorem. In thi...

Hindman’s Theorem says that every finite coloring of the positive natural numbers has a monochromatic set of finite sums. Ramsey algebras, recently introduced, are structures that satisfy an analogue ...

Given subset E of natural numbers FS(E) is defined as the collection of all sums of elements of finite subsets of E and any translation of FS(E) is said to be Hilbert cube. We can define the multiplic...

Generalisations du theoreme de Gahvin-Prikry aux grands cardinaux. Etude de quelques proprietes de Ramsey duales pour les cardinaux non denombrables

For f ∈ `∞(N) let Tf be defined by Tf(n) = 1 n ∑n i=1 f(i). We investigate permutations g of N, which satisfy Tf(n)−Tfg(n)→ 0 as n→∞ with fg(n) = f(gn) for f ∈ `∞(N) (i.e. g is in the Lévy group G), o...