### A game on partial orderings

by Fuchino and Koppelberg and Shelah. [FKSh:556]

Topology and its Applications, 1996

We study the determinacy of the game G_kappa (A)
introduced in [FKSh:549] for uncountable regular kappa and
several classes of partial orderings A . Among trees or Boolean
algebras, we can always find an A such that G_kappa (A) is
undetermined. For the class of linear orders, the existence of
such A depends on the size of kappa^{< kappa} . In
particular we obtain a characterization of
kappa^{< kappa}= kappa in terms of determinacy of the game
G_kappa (L) for linear orders L .

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