Ideal Games and Ramsey Sets

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 Sets, the Ramsey Ideal, and Other Classes over R

Determinacy of Adversarial Gowers Games

We prove a game theoretic dichotomy for Gδσ sets of block sequences in vector spaces that extends, on the one hand, the block Ramsey theorem of W. T. Gowers proved for analytic sets of block sequences and, on the other hand, M. Davis’ proof of Σ3 determinacy.

Strolling through Paradise

With each of the classical tree–like forcings adjoining a new real, one can associate a σ– ideal on the reals in a natural way. For example, the ideal of Marczewski null sets s corresponds to Sacks forcing S, while the ideal of nowhere Ramsey sets r corresponds to Mathias forcing R. We show (in ZFC) that none of these ideals is included in any of the others. We also discuss Mycielski’s ideal P2...

F U N D a M E N T a Mathematicae Strolling through Paradise

With each of the classical tree-like forcings adjoining a new real, one can associate a σ-ideal on the reals in a natural way. For example, the ideal s0 of Marczewski null sets corresponds to Sacks forcing S, while the ideal r0 of nowhere Ramsey sets corresponds to Mathias forcing R. We show (in ZFC) that none of these ideals is included in any of the others. We also discuss Mycielski’s ideal P...