Cardinals of the $P_κ(λ)$-Filter Games

Tom Benhamou; Victoria Gitman

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arXiv · 2504.00119

Cardinals of the $P_κ(λ)$-Filter Games

Tom Benhamou·Victoria Gitman

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Abstract

We investigate forms of filter extension properties in the two-cardinal setting involving filters on $P_κ(λ)$. We generalize the filter games introduced by Holy and Schlicht in \cite{HolySchlicht:HierarchyRamseyLikeCardinals} to filters on $P_κ(λ)$ and show that the existence of a winning strategy for Player II in a game of a certain length can be used to characterize several large cardinal notions such as: $λ$-super/strongly compact cardinals, $λ$-completely ineffable cardinals, nearly $λ$-super/strongly compact cardinals, and various notions of generic super and strong compactness. We generalize a result of Nielson from \cite{NielsenWelch:games_and_Ramsey-like_cardinals} connecting the existence of a winning strategy for Player II in a game of finite length and two-cardinal indescribability. We generalize the result of \cite{ForMagZem} to construct a fine $κ$-complete precipitous ideal on $P_κ(λ)$ from a winning strategy for Player II in a game of length $ω$. Finally, we improve Theorems 1.2 and 1.4 from \cite{ForMagZem} and partially answer questions Q.1 and Q.2 from \cite{ForMagZem}.

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BibTeX RIS

Tom Benhamou, Victoria Gitman. 2026-02-18. Cardinals of the $P_κ(λ)$-Filter Games. https://arxiv.org/abs/2504.00119

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