Narrow Results

We answer a question of Slaman and Steel by showing that a version of Martin’s conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin’s conjecture, consists of showing that we can always reduce to the case of a continuous function.

We show that in the ${\mathbb{P}}_{max}$ extension of a certain Chang-type model of determinacy, if $\kappa \in \{{\omega }_1,{\omega }_2,{\omega }_3\}$, then the restriction of the club filter on $\kappa \cap$ Cof$(\omega )$ to HOD is an ultrafilter in HOD. This answers Question 4.11 of [O. Ben-Neria and Y. Hayut, On $\omega$-strongly measurable cardinals, Forum Math. Sigma11 (2023) e19].

We study the notion of ${𝒥}$-MAD families where ${𝒥}$ is a Borel ideal on $\omega$. We show that if ${𝒥}$ is any finite or countably iterated Fubini product of the ideal of finite sets $\mathrm{\text{Fin}}$, then there are no analytic infinite ${𝒥}$-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective ${𝒥}$-MAD families; and under the full Axiom of Determinacy $+$$V=L(ℝ)$ or under ${AD}^{+}$ there are no infinite ${𝒥}$-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal $\mathrm{\text{Fin}}$, which corresponds to the classical notion of MAD families, as well as to the ideal $\mathrm{\text{Fin}}\otimes \mathrm{\text{Fin}}$. The proofs combine ideas from invariant descriptive set theory and forcing.

We introduce the notion of $(\mathrm{\Gamma },E)$-determinacy for $\mathrm{\Gamma }$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where $S=2=\{0,1\}$ or $S=\omega$. We show that for all shift actions by countable groups $G$, and any “reasonable” pointclass $\mathrm{\Gamma }$, that $(\mathrm{\Gamma },E_G)$-determinacy implies $\mathrm{\Gamma }$-determinacy. We also prove a corresponding result when $E$ is a subshift of finite type of the shift map on $2^{\mathbb{Z}}$.

Assume $ZF+AD$. If $𝜖$ is an ordinal and X is a set of ordinals, then ${[X]}_{\ast }^{𝜖}$ is the collection of order-preserving functions $f:𝜖\to X$ which have uniform cofinality $\omega$ and discontinuous everywhere. The weak partition properties on ${\omega }_1$ and ${\omega }_2$ yield partition measures on ${[{\omega }_1]}_{\ast }^{𝜖}$ when $𝜖<{\omega }_1$ and ${[{\omega }_2]}_{\ast }^{𝜖}$ when $𝜖<{\omega }_2$. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every $𝜖<{\omega }_1$ and function $\mathrm{\Phi }:{[{\omega }_1]}^{𝜖}\to {\omega }_1$, there is a club $C\subseteq {\omega }_1$ and a $\zeta <𝜖$ so that for all $f,g\in {[C]}_{\ast }^{𝜖}$, if $f\upharpoonright \zeta =g\upharpoonright \zeta$ and $sup(f)=sup(g)$, then $\mathrm{\Phi }(f)=\mathrm{\Phi }(g)$. For every $𝜖<{\omega }_2$ and function $\mathrm{\Phi }:{[{\omega }_2]}^{𝜖}\to {\omega }_2$, there is an $\omega$-club $C\subseteq {\omega }_2$ and a $\zeta <𝜖$ so that for all $f,g\in {[C]}_{\ast }^{𝜖}$, if $f\upharpoonright \zeta =g\upharpoonright \zeta$ and $sup(f)=sup(g)$, then $\mathrm{\Phi }(f)=\mathrm{\Phi }(g)$. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of $\mathcal{𝒫}({\omega }_2):|{[{\omega }_1]}^{\omega }|<|{[{\omega }_1]}^{<{\omega }_1}|$. $|{[{\omega }_2]}^{\omega }|<|{[{\omega }_2]}^{<{\omega }_1}|<|{[{\omega }_2]}^{{\omega }_1}|<|{[{\omega }_2]}^{<{\omega }_2}|$. $\neg (|{[{\omega }_1]}^{<{\omega }_1}|\le |{[{\omega }_2]}^{\omega }|)$. $\neg (|{[{\omega }_1]}^{{\omega }_1}|\le |{[{\omega }_2]}^{<{\omega }_1}|)$. It will also be shown that ${[{\omega }_1]}^{\omega }$ has the Jónsson property: For every $\mathrm{\Phi }:{}^{<\omega }({[{\omega }_1]}^{\omega })\to {[{\omega }_1]}^{\omega }$, there is an $X\subseteq {[{\omega }_1]}^{\omega }$ with $|X|=|{[{\omega }_1]}^{\omega }|$ so that $\mathrm{\Phi }[{}^{<\omega }X]\ne {[{\omega }_1]}^{\omega }$.