Rocky Mountain Journal of Mathematics

Almost compatible functions and infinite length games

Steven Clontz and Alan Dow

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${\mathcal{A}}'(\kappa)$ asserts the existence of pairwise almost compatible finite-to-one functions $A\to \omega$ for each countable subset $A$ of $\kappa$. The existence of winning $2$-Markov strategies in several infinite-length games, including the Menger game on the one-point Lindelofication $\kappa^\dagger$ of $\kappa$, are guaranteed by ${\mathcal{A}}'(\kappa)$. ${\mathcal{A}}'(\kappa)$ is implied by the existence of cofinal Kurepa families of size $\kappa$, and thus, holds for all cardinals less than $\aleph _\omega$. It is consistent that ${\mathcal{A}}'({\aleph _\omega })$ fails; however, there must always be a winning $2$-Markov strategy for the second player in the Menger game on $\omega_\omega^\dagger$.

Article information

Rocky Mountain J. Math., Volume 48, Number 2 (2018), 463-483.

First available in Project Euclid: 4 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 91A44: Games involving topology or set theory
Secondary: 03E35: Consistency and independence results 03E55: Large cardinals

Selection games almost compatible functions covering properties


Clontz, Steven; Dow, Alan. Almost compatible functions and infinite length games. Rocky Mountain J. Math. 48 (2018), no. 2, 463--483. doi:10.1216/RMJ-2018-48-2-463.

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