The generalized continuum hypothesis above a strongly compact cardinal

In this paper, we study how the classification complexity of natural orbit equivalence relations changes when the full homeomorphism group of a compact metrizable space is replaced by a dense non-closed subgroup. For a compact space $X$ and a subgroup $G \leq \mathcal{H}(X)$, we consider three canonical actions: the left shift action on $\mathcal{H}(X)$, the induced hyperspace action on $\mathcal{F}(X)$, and the conjugation action on $G$ We first analyze subgroups of the group $\mathcal{H}^+([0,1])$ of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.

math.LO↗

arXiv2026

Stationary common-neighborhood properties and partition hypotheses

We use stationary common-neighborhood properties to study highly connected Ramsey relations and partition hypotheses. For weakly compact $κ$, $\operatorname{Coll}(ω_1,{<}κ)$ forces $ω_2\to_{\mathrm{hc},<5}(ω_2)^2_ω$ and $\operatorname{PH}_1(ω_2)$. If $κ$ is $T^{κ^+}_{ω_1}$-Ramsey, the same collapse forces that every countable coloring of $[ω_2]^2$ has a stationary set $X\subseteqω_2$ and a color $i$ such that every finite subset of $X$ has stationarily many color-$i$ common neighbors in $X$. From one weakly compact cardinal, we obtain a model of the ${<}5$-edge relation at $ω_3$ and $\operatorname{PH}_1(ω_3)$, in which $\check H^2(ω_3,A_d)\ne0$ for every nontrivial abelian group $A$. This separates $\operatorname{PH}_1(ω_3)$ from $\operatorname{PH}_2(ω_3)$, with the exact consistency strength of one weakly compact cardinal. We also show that $\operatorname{PH}_1(ω_2\timesω_5)$ is equiconsistent with two weakly compact cardinals.

math.LO↗

arXiv2026

An entire function that violates quasiminimality

Zilber's celebrated quasiminimality conjecture states that a subset of $\mathbb C$ defined by polynomials and exponentials is either countable, or has countable complement. Koiran has asked whether the same could be true replacing $\exp$ with any unary entire function. We give a negative answer to Koiran's question, using results from the theory of holomorphic approximation.

math.LO↗