On completely Ramsey sets

Coding into Ramsey Sets

In [6] W. T. Gowers formulated and proved a Ramseytype result which lies at the heart of his famous dichotomy for Banach spaces. He defines the notion of weakly Ramsey set of block sequences of an infinite dimensional Banach space and shows that every analytic set of block sequences is weakly Ramsey. We show here that Gowers’ result follows quite directly from the fact that all Gδ sets are weak...

Completely reducible sets

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a resu...

Ideal Games and Ramsey Sets

It is shown that Matet’s characterization ([9]) of the Ramsey property relative to a selective co-ideal H, in terms of games of Kastanas ([6]), still holds if we consider semiselectivity ([3]) instead of selectivity. Moreover, we prove that a co-ideal H is semiselective if and only if Matet’s game-theoretic characterization of the H-Ramsey property holds. This lifts Kastanas’s characterization ...

Ramsey Sets, the Ramsey Ideal, and Other Classes over R

On Ramsey Numbers for Sets Free of Prescribed Differences

For a positive integer d, a set S of positive integers is difference d-free if |x−y| 6 = d for all x, y ∈ S. We consider the following Ramseytheoretical question: Given d, k, r ∈ Z, what is the smallest integer n such that every r-coloring of [1, n] contains a monochromatic kelement difference d-free set? We provide a formula for this n. We then consider the more general problem where the monoc...