arXiv subjects

David J. Fernández Bretón

Publications and source records attributed to David J. Fernández Bretón.

3 recordsLinked to original sources

arXiv2026

Hindman and Owings-like theorems without the Axiom of Choice

We investigate Hindman- and Owings-type Ramsey-theoretic statements in Zermelo-Fraenkel set theory without the Axiom of Choice, with some occasional extra assumptions (such as the Axiom of Dependent Choice and/or the Axiom of Determinacy). We study several variations of Hindman's theorem on $\mathbb Q$-vector spaces; notably, we show that the uncountable analog of Hindman's theorem fails for the additive group of $\mathbb R$ (under ZF), and for $\mathbb Q$-vector spaces of uncountable dimension (under DC if such dimension is not well-orderable), among other results. In contrast, for Owings-type configurations, we obtain several positive results, especially when assuming AD. These results highlight the interaction between determinacy, algebraic structure, and dimension in the study of infinite Ramsey theory without the Axiom of Choice.

math.LO↗

arXiv2015

Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

We answer two questions of Hindman, Steprāns and Strauss, namely we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases: we also construct (assuming Martin's Axiom for countable partial orders, i.e. $\mathrm{cov}(\mathcal M)=\mathfrak c$), on the Boolean group, a strongly summable ultrafilter that is not additively isomorphic to any union ultrafilter.

math.LO↗

arXiv2013

Every strongly summable ultrafilter on $\bigoplus\mathbb Z_2$ is sparse

We investigate the possibility of the existence of nonsparse strongly summable ultrafilters on certain abelian groups. In particular, we show that every strongly summable ultrafilter on the countably infinite Boolean group is sparse. This answers a question of Hindman, Steprāns and Strauss.

math.LO↗