Uncountable sets and an infinite linear order game

TL;DR

This paper explores an infinite game on linear orders, demonstrating that Player II can have a winning strategy on uncountable sets in certain dense linear orders, extending previous results about countable sets.

Contribution

It constructs dense linear orders of any infinite size where Player II wins on all payoff sets, generalizing earlier results from countable to uncountable cases.

Findings

Player II has a winning strategy on uncountable sets in certain dense linear orders.

Construction of dense linear orders of any infinite size with specific game-winning properties.

Highlights the complexity of extending countable set characterizations to uncountable linear orders.

Abstract

An infinite game on the set of real numbers appeared in Matthew Baker's work [Math. Mag. 80 (2007), no. 5, pp. 377--380] 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.