Ramsey games

A. Hajnal, Zs. Nagy
1984 Transactions of the American Mathematical Society  
The paper deals with game-theoretic versions of the partition relations a -» (ß)^T and a -> (ß)^ introduced in [2]. The main results are summarized in the Introduction. 0. Introduction. In their paper [2], Baumgartner, Galvin, McKenzie and Laver introduced a new game. Let a,ß be ordinals, r a cardinal. The Ramsey game Z?(a, r, ß) is defined as follows. There are two players, White and Black, who alternately pick previously unchosen members of [a]T. At limit stages it is of course White's turn
more » ... urse White's turn to move. The game ends when the set [a]T is exhausted. White wins if there is a set A C a, tp A = ß with [A]T C W', where W C [a]T is the set of r element sets which White chose; otherwise Black wins. We say that White (Black) wins R(a,T,ß) if White (Black) has a winning strategy. This game can be considered as a game-theoretic version of the partition relation a -► (ß)2. The game R(a, < r,ß) is defined similarly, where the choices are made from [o]) for r < ui. However the proof given there yields the following THEOREM [2]. Assume k > w, 0 < r < w and k -> (ß)2r-i ■ Then White wins R(tz, /c+ and Black still wins R(k+, 2, k+) but we have no other information. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 816 A. HAJNAL AND ZS. NAGY For 2 < r = r < u we have even less information and we only state the simplest case of our results. Using a result of R. Laver we can prove (2) Assume 2H° = Ni and the negation of Chang's conjecture holds, i.e. #2 -/+
doi:10.1090/s0002-9947-1984-0743746-3 fatcat:vxh5vy2wonfeveddahwy4qiz7q