Games and the Silver property
Vladimir Grigor'evich Kanovei, Marcin Sabok, Jindřich Zapletal · Cambridge University Press eBooks · 2013
Integer games connected with σ-ideals Many of the σ-ideals considered in this book have integer games associated with them. As a result, they satisfy several interconnected properties, among them the selection property that will be instrumental in upgrading the canonization results to Silver-style dichotomies for these σ-ideals. The subject of integer games and σ-ideals was treated in Zapletal (2008) rather extensively, but on a case-by-case basis. In this section, we provide a general framework, show that it is closely connected with uniformization theorems, and prove a couple of dichotomies under the assumption of the Axiom of Determinacy. To help motivate the following definitions, we will consider a simple task. Let I be a collection of subsets of a Polish space X , closed under subsets. Suppose that I has a basis , a Borel set B ⊂ ω ω × X such that a subset of X is in I iff it is covered by a vertical section of B . Let A ⊂ X be a set, and consider an infinite game in which Player I produces a point y ∈ ω ω and Player II a point x ∈ X . Player II wins if x ∈ A \ B y . Certainly, if A ∈ I then Player I has a winning strategy that completely disregards moves of Player II – just producing the vertical section of the basis which covers the set A .