arXiv subjects

Carlos López-Callejas

Publications and source records attributed to Carlos López-Callejas.

4 recordsLinked to original sources

arXiv2026

A rank function for Fra\"ıssé classes and the rank property

Given a hereditary class $\mathcal{F}$ of finite relational structures, the rank function $\mathsf{rk}:σ\mathcal{F}\toω_1\cup\{\infty\}$, introduced by Kubiś 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\"ıssé 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 $X\inσ\mathcal{F}$. 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 $ω^{β_1}\cdot c_1$ is the leading Cantor normal form term of $α\geqω$, then $\mathsf{rk}(α)=ω\cdotβ_1+\lfloor\log_2 c_1\rfloor$.

math.LO↗

arXiv2026

Diamonds on trees

We generalize the diamond principle and its variants using the notion of stationarity in trees introduced by Brodsky in [Brodsky, A. M., A theory of stationary trees and the balanced Baumgartner--Hajnal--Todorcevic theorem for trees. The Bulletin of Symbolic Logic]. In particular, we show that if $T$ is a nonspecial $ω_1$-tree, then $\diamondsuit_T \implies \diamondsuit$, and if $T$ is a Suslin tree, then $\diamondsuit_T \iff \diamondsuit$. We also prove that $\diamondsuit^*$ implies $\diamondsuit_T$ (yielding the consistency of $\diamondsuit_T$) and establish the consistency of $\neg\diamondsuit^* + (\forall T\text{ nonspecial }ω_1\text{-tree }(\diamondsuit_T))$. Finally, we demonstrate that it is consistent with $\diamondsuit$ that there exists a nonspecial $ω_1$-tree with $(\neg\diamondsuit_T)$, introducing two forcing properties -- $σ(S)$-closed and strategically closed in models -- which are preserved under countable support iterations.

math.LO↗

arXiv2023

High dimensional sequential compactness

We give examples of $n$-sequentially compact spaces that are not $(n+1)$-sequentially compact under several assumptions. We improve results from Kubis and Szeptycki by building such examples from $\mathfrak{b=c}$ and $\diamondsuit(\mathfrak{b})+\mathfrak{d}=ω_1$. We also introduce a new splitting-like cardinal invariant and then show that the same holds under $\mathfrak{s=b}$.

math.GN↗

arXiv2023

Generalized independence

We explore different generalizations of the classical concept of independent families on $ω$ following the study initiated by Fisher and Montoya. We show that under $\diamondsuit^*(κ)$ we can get strongly independent families on $κ$ of size $2^κ$ and present an equivalence of $\mathsf{GCH}$ in terms of strongly independent families. We merge the two natural ways of generalizing independent families through a filter or an ideal and we focus on the $\mathcal{C}$-independent families, where $\mathcal{C}$ is the club filter. Also we show a relationship between the existence of $\mathcal{J}$-independent families and the saturation of the ideal $\mathcal{J}$.

math.LO↗