Meager-nowhere dense games. VI: Markov \(k\)-tactics (Q1922052)

[For part V see the author, Quaest. Math. 17, No. 4, 419-435 (1994; Zbl 0820.90147).] Let \(X\) be a \(T_1\)-space with no isolated points. Players A and B play the following game: They play an inning per positive integer. In the \(n\)th inning A chooses a first category subset \(O_n\), and B responds with a nowhere dense subset \(T_n\). A must further obey the rule that for each \(n O_n\) is a proper subset of \(O_{n+1}\). A play \(O_1,T_1,\dots,O_n\), \(T_n,\dots\) is won by B if \(\bigcup^\infty_{n=1} O_n\subseteq \bigcup^\infty_{n=1} T_n\). If B can in each inning remember all the preceding moves made by A, then B has a winning strategy. To what extent does B need so much memory to ensure a win? This question has been fairly extensively studied. In the paper under review, two sorts of limitations on B's memory are considered. For a fixed positive integer \(k\), a \(k\)-tactic for B is a strategy which uses as information no more than at most the \(k\) most recent moves of A; a Markov \(k\)-tactic uses besides this information also the number of the inning in progress. In this paper, some of the basic combinatorial circumstances under which a winning \(k\)-tactic (Theorems 16 and 18) or a winning Markov \(k\)-tactic (Theorem 12) for B exists are identified and applied to specific examples.