Combinatorics in Higher Solovay Models

Abstract

We construe the singular-cardinal analogue of the classical Solovay model. Starting with large cardinal assumptions in the realm of supercompactness, we show that the our inner model captures a substantial portion of the combinatorics of $L(\mathcal{P}(κ))$ that are typically implied by Woodin&#39;s axiom $I_0$. Among other things, we show that in our higher Solovay model there are no $κ^+$-sequences of distinct members of $\mathcal{P}(κ)$ and that Shelah&#39;s approachability property $\AP_κ$ fails. We prove that every set in our inner model satisfies a singular analogue of the complete Ramsey property and that the partition relation $κ\xrightarrow[]{\mathrm{OD}} (ω)^ω_{V_μ}$ holds for all $μ<κ$.