Abstract
${\mathcal{A}}'(\kappa)$ 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 $\omega_\omega^\dagger$.
Citation
Steven Clontz. Alan Dow. "Almost compatible functions and infinite length games." Rocky Mountain J. Math. 48 (2) 463 - 483, 2018. https://doi.org/10.1216/RMJ-2018-48-2-463
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