Variations on a game of Gale (III): remainder strategies

Abstract

An infinite set X is given. D. Gale, in correspondence with J. Mycielski, described the following game in which players one and two play an inning per positive integer: In the nth inning one chooses a finite subset Xn of X, and two chooses a point xn from (X1∪ … ∪Xn)\{x1,…,xn−1}. A playis won by two if . Gale asked whether two could have a winning strategy which depends for each n on knowledge of only the contents of the setIn mathematical terms, is there a function F defined on the collection of finite subsets of X such that:for every sequence X1, x1, …, Xn, xn,…. where each Xn is a finite subsetof X and for each nwe have We shall call a strategy of this sort a remainder strategy for two. If there is some finite subset U of X such that F(U) ∉ U, then F cannot be a winning remainder strategy for two, because one can defeat F by choosing U each inning. So, when studying remainder strategies for two we may as well assume that for each finite set U ⊂ X, F(U) ∈ U.