arXiv subjects

David José Fernández-Bretón

Publications and source records attributed to David José Fernández-Bretón.

2 recordsLinked to original sources

arXiv2026

$\mathbb Z$-Ramsey ultrafilters

We study $\mathbb{Z}$-Ramsey ultrafilters, ultrafilters containing witnesses to every shift-invariant instance of Ramsey's theorem. We prove that it is consistent that there are no $\mathbb Z$-Ramsey ultrafilters. We also prove that every $(\mathbb{Z},3)$-Ramsey ultrafilter, as well as every $\mathbb{Z}$-Ramsey P-point, is selective. Further, we exhibit a generic extension -- using quotient algebras of the form $\mathcal{P}(\mathbb{Z})/\mathcal{I}$ for certain $F_{σ}$-ideals -- that contains P-points that are not $\mathbb{Z}$-Ramsey ultrafilters, thereby addressing open questions raised by Petrenko and Protasov.

math.LO↗

arXiv2019

Stable ordered union ultrafilters and $\mathrm{cov}(\mathcal{M})<\mathfrak c$

A union ultrafilter is an ultrafilter over the finite subsets of $ω$ that has a base of sets of the form $\mathrm{FU}(X)$, where $X$ is an infinite pairwise disjoint family and $\mathrm{FU}(X)=\{\bigcup F\big|F\in[X]^{<ω}\setminus\{\varnothing\}\}$. The existence of these ultrafilters is not provable from the $\mathsf{ZFC}$ axioms, but is known to follow from the assumption that $\mathrm{cov}(\mathcal{M})=\mathfrak c$. In this article we obtain various models of $\mathsf{ZFC}$ that satisfy the existence of union ultrafilters while at the same time $\mathrm{cov}(\mathcal{M})<\mathfrak c$.

math.LO↗