### A characterization of determinacy for Turing degree games

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We define a new large cardinal axiom that fits between ${A}_{3}$ and ${A}_{4}$ in the hierarchy of axioms described in [SRK]. We use this new axiom to obtain a Laver sequence for extendible cardinals, improving the known large cardinal upper bound for the existence of such sequences.

Assuming large cardinals, we show that every κ-complete filter can be generically extended to a V-ultrafilter with well-founded ultrapower. We then apply this to answer a question of Abe.

If κ < λ are such that κ is both supercompact and indestructible under κ-directed closed forcing which is also (κ⁺,∞)-distributive and λ is ${2}^{\lambda}$ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], δ < κ | δ is δ⁺ strongly compact yet δ is not δ⁺ supercompact must be unbounded in κ. We show that the large cardinal hypothesis on λ is necessary by constructing a model containing a supercompact cardinal κ in which no cardinal δ > κ is ${2}^{\delta}=\delta \u207a$ supercompact, κ’s supercompactness...

We construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal ĸ is ${\u0138}^{+}$ supercompact and in which every strongly compact cardinal has its strong compactness resurrectible.

We show how to reduce the assumptions in consistency strength used to prove several theorems on universal indestructibility.

We study pairs (V, V₁), V ⊆ V₁, of models of ZFC such that adding κ-many Cohen reals over V₁ adds λ-many Cohen reals over V for some λ > κ.