Ideal games and Ramsey sets

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

A Ramsey bound on stable sets in Jordan pillage games

Jordan [2006] defined ‘pillage games’, a class of cooperative games whose dominance operator is represented by a ‘power function’ satisfying coalitional and resource monotonicity axioms. In this environment, he proved that stable sets must be finite. We provide a graph theoretical interpretation of the problem which tightens the finite bound to a Ramsey number. We also prove that the Jordan pil...

Ramsey games with giants

The classical result in the theory of random graphs, proved by Erdős and Rényi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of r colors. The goal can be either to create a giant component in e...

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...