arXiv subjects

Carlos A. Di Prisco

Publications and source records attributed to Carlos A. Di Prisco.

3 recordsLinked to original sources

arXiv2026

Combinatorics of Ramsey ideals

We primarily study several combinatorial properties of Ramsey-type ideals on countably infinite sets. Specifically, we show new combinatorial characterizations of Ramsey ideals through various partition and convergence properties. Furthermore, we analyze ideal versions of some relevant high-dimensional Ramsey-type theorems, in order to research ideals related to finite colorings of barriers on the natural numbers as well as ideals associated with finite partitions of any family of finite subsets of the natural numbers. In particular, Galvin ideals are introduced as an intermediate combinatorial concept between Ramsey ideals and semiselective ideals.

math.LO↗

arXiv2026

Some combinatorial properties of semiselective ideals

We present several combinatorial properties of semiselective ideals on the set of natural numbers. The continuum hypothesis implies that the complement of every selective ideal contains a selective ultrafilter, however for semiselective ideals this is not the case. We prove that under certain hypothesis, for instance $V=L$, there are semiselective ideals whose complement does not contain a selective ultrafilter, and that it is also consistent that the complement of every semiselective ideal contains a selective ultrafilter; specifically, we show that if $V=L$ then there is a generic extension of $V$ where this occurs. We present other results concerning semiselective ideals, namely an alternative proof of Ellentuck's theorem for the local Ramsey property, and we prove some facts about the additivity of the ideal of local Ramsey null sets, and also about the generalized Suslin operation on the algebra of local Ramsey sets.

math.LO↗

arXiv2025

Topological games in Ramsey spaces

Topological Ramsey theory studies a class of combinatorial topological spaces, known as topological Ramsey spaces, unifying the essential features of those combinatorial frames where the Ramsey property is equivalent to the Baire property. In this article, we present a general overview of the combinatorial structure of topological Ramsey spaces and their main properties, and we propose an alternative proof of the abstract Ellentuck theorem for a large family of axiomatized topological Ramsey spaces. Additionally, we introduce the notion of selective axiomatized topological Ramsey space, and generalize Kastanas games in order to characterize the Ramsey property for this broad family of topological Ramsey spaces through topological games.

math.LO↗