Uncountable sets and an infinite linear order game
Keywords:
Baker Game, infinite-length games, strategiesAbstract
An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377380] in which he asks whether it can help characterize countable subsets of the reals. This question is in a similar spirit to how the Banach-Mazur Game characterizes meager sets in an arbitrary topological space.
In a recent paper, Will Brian and Steven Clontz prove that in Baker's game, Player II has a winning strategy if and only if the payoff set is countable. They also asked if it is possible, in general linear orders, for Player II to have a winning strategy on some uncountable set.
To this we give a positive answer and moreover construct, for every infinite cardinal $\kappa$, a dense linear order of size $\kappa$ on which Player II has a winning strategy on all payoff sets. We finish with some future research questions, further underlining the difficulty in generalizing the characterization of Brian and Clontz to linear orders.
References
Dzmitry Badziahin, Stephen Harrap, Erez Nesharim, and David Simmons, Schmidt games and Cantor winning sets, arXiv:1804.06499v2 (2020).
Matthew H. Baker, Uncountable sets and an infinite real number game, arXiv:math/0606253v1 (2006).
Magnus D. Ladue, The Cantor Game: winning strategies and determinacy, arXiv:1701.09087v1 (2017).
Thomas Jech, Set Theory: The Third Millenium Edition, revised and expanded, Springer-Verlag, New York, 2003.
Walter Rudin, Principles of Mathematical Analysis (3rd ed.), McGraw-Hill, New York, 1976.
John C. Oxtoby, Measure and Category (2nd ed.), Springer-Verlag, 1980.
Alexander S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, 1995.
Will Brian and Steven Clontz, Elementary Submodels, Coding Strategies, and an Infinite Real Number Game, Topology Proceedings 62 (2023), 151-161.
Stevo Todorčević, Partition relations for partially ordered sets, Acta Math. 155 (1985).
Stevo Todorčević, Aronszajn orderings, Publications de L'Institut Mathematique, Nouvelle serie 57 (1995), 29-46.
Stevo Todorčević, CHAPTER 6 - Trees and Linearly Ordered Sets, in: Kenneth Kunen and Jerry E. Vaughan (eds.), Handbook of Set-Theoretic Topology, North-Holland, Amsterdam, 1984, 235-293.
