Perfect set dichotomy theorem in generalized Solovay model

Abstract

We prove that the perfect set dichotomy theorem holds in the Solovay model $V ((ω^ω)^{V[G]})$. Namely, for every equivalence relation $E$ on $\mathbb{R}$, either $\mathbb{R}/E$ is well-orderable or there exists a perfect set consisting of $E$-inequivalent reals. Furthermore we consider a generalization of the Solovay model for an uncountable regular cardinal $μ$ and show the perfect set dichotomy theorem for $μ^μ$ also holds in that model. We establish the three element basis theorem for uncountable linear orders in the Solovay model for a weakly compact cardinal, in a general form covering the uncountable case.