Jareb Navarro-Castillo

arXiv2026

A rank function for Fra\"{\i}ss\'{e} classes and the rank property

Given a hereditary class $\mathcal{F}$ of finite relational structures, the rank function $\mathsf{rk}:\sigma\mathcal{F}\to\omega_1\cup\{\infty\}$, introduced by Kubi\'{s} and Shelah, measures how far a countable structure is from being universal within its class: $\mathsf{rk}(X)=\infty$ if and only if the Fra\"{\i}ss\'{e} limit embeds into $X$. We say that $\mathcal{F}$ has the Rank Property (RP) if every countable ordinal is realized as the rank of some $X\in\sigma\mathcal{F}$. We develop the basic theory of the rank function and establish RP for three families of classes: those satisfying the free amalgamation property and the full extension property (covering graphs, hypergraphs, and many others); finite tournaments; and finite linear orders. For the latter, we compute the rank of every countable ordinal: if $\omega^{\beta_1}\cdot c_1$ is the leading Cantor normal form term of $\alpha\geq\omega$, then $\mathsf{rk}(\alpha)=\omega\cdot\beta_1+\lfloor\log_2 c_1\rfloor$.

math.LO↗

arXiv2025

Q-points, selective ultrafilters, and idempotents, with an application to choiceless set theory

We study ultrafilters from the perspective of the algebra in the Čech-Stone compactification of the natural numbers, and idempotent elements therein. The first two results that we prove establish that, if $p$ is a Q-point (resp. a selective ultrafilter) and $\mathscr F^p$ (resp. $\mathscr G^p$) is the smallest family containing $p$ and closed under iterated sums (resp. closed under Blass--Frol\'ık sums and Rudin--Keisler images), then $\mathscr F^p$ (resp. $\mathscr G^p$) contains no idempotent elements. The second of these results about a selective ultrafilter has the following interesting consequence: assuming a conjecture of Blass, in models of the form $\mathbf{L}(\mathbb R)[p]$ where $\mathbf{L}(\mathbb R)$ is a Solovay model (of $\mathsf{ZF}$ without choice) and $p$ is a selective ultrafilter, there are no idempotent elements. In particular, the theory $\mathsf{ZF}$ plus the existence of a nonprincipal ultrafilter on $ω$ does not imply the existence of idempotent ultrafilters, which answers a question of DiNasso and Tachtsis (Proc. Amer. Math. Soc. 146, 397-411). Following the line of obtaining independence results in $\mathsf{ZF}$, we finish the paper by proving that $\mathsf{ZF}$ plus "every additive filter can be extended to an idempotent ultrafilter" does not imply the Ultrafilter Theorem over $\mathbb R$, answering another question of DiNasso and Tachtsis from the same paper.

math.LO↗