arXiv subjects

Tan Özalp

Publications and source records attributed to Tan Özalp.

6 recordsLinked to original sources

arXiv2026

Canonical equivalence relations on $\mathrm{FIN}^{[\infty]}_2$

Answering a question of Todorcevic, we prove higher-dimensional canonization theorems for the topological Ramsey space $\mathrm{FIN}^{[\infty]}_2$. Our results build upon the work of Lopez-Abad and continue the line of research initiated by Erdős and Rado, and further developed by Pudlák and Rödl, Prömel and Voigt, Taylor, and Klein and Spinas. We identify the canonical functions on fronts of $\mathrm{FIN}^{[\infty]}_2$ and develop an extension of the separating-mixing technique of Prömel and Voigt to canonize arbitrary functions $g:\mathcal{F}\toω$, where $\mathcal{F}$ is a front of $\mathrm{FIN}^{[\infty]}_2$. We further extend our canonization theorem to arbitrary Borel maps $g:\mathrm{FIN}^{[\infty]}_2\to\mathbb{R}$, establishing new canonical Ramsey theorems for Polish spaces. In particular, we provide a complete classification of the canonical functions on $\mathrm{FIN}^{[n]}_2$, together with an explicit formula for their number as a function of $n$.

math.LO↗

arXiv2026

On the number of $Q$-points

We show that, up to isomorphism, the number of $Q$-points is either finite, $2^{\mathfrak{d}}$ or $2^{\mathfrak{c}}$. This answers a question asked by Borodulin-Nadzieja, Martínez-Celis, Morawski and Świerczyńska, and by Halbeisen and the authors. We also show that under mild hypotheses, the existence of infinitely many $Q$-points implies the existence of non-atomic $Q$-measures, and of $2^{\mathfrak{c}}$-many Tukey-top $Q$-points, strengthening results of Raghavan and of Borodulin-Nadzieja et al..

math.LO↗

arXiv2026

Laver ultrafilters

We introduce $\textit{Laver ultrafilters}$, namely ultrafilters $\mathcal{U}$ for which the associated Laver forcing $\mathbb{L}_{\mathcal{U}}$ has the Laver property. We give simple combinatorial characterisations of these ultrafilters, which allow us to analyse their position among several well-studied combinatorial classes, including $P$-points, rapid ultrafilters, and ultrafilters arising in Baumgartner's $\mathcal{I}$-ultrafilter framework. In particular, we show that the class of Laver ultrafilters properly contains the class of rapid $P$-points and that it is properly contained both in the class of hereditarily rapid- and in the class of measure zero ultrafilters. Finally, we investigate the (generic) existence of Laver ultrafilters and establish bounds on their generic existence number. In particular, we show that it is consistent that $P$-points do not exist while Laver ultrafilters exist generically.

math.LO↗

arXiv2025

Tukey-idempotency and strong p-points

We characterize strong $p$-point ultrafilters by showing that they are exactly those $p$-points that are not Tukey above $(ω^ω,\leq)$; or equivalently, those $p$-points that are not Tukey-idempotent. Moreover, we show that there are no Canjar ultrafilters on measurable cardinals. We make use of tools which were motivated by topological Ramsey spaces, developed in \cite{Benhamou/Dobrinen24}, and furthermore, show that ultrafilters arising from most of the known topological Ramsey spaces are Tukey-idempotent. Our results answer questions of Hrušák and Verner \cite[Question 5.7]{Hrusak/Verner11}, Brook-Taylor \cite[Question 3.6]{QuestionGeneralized}, and partially Benhamou and Dobrinen \cite[Question 5.6]{Benhamou/Dobrinen24}.

math.LO↗

arXiv2025

There may be exactly $n$ $Q$-points

We generalize the main result of arXiv:2505.17960 and show the consistency of the statement ``There are exactly $n$ $Q$-points up to isomorphism" for any finite $n$. Furthermore, we show that the above statement for $n=2$ can alternatively be obtained by a length-$ω_2$ countable support iteration of Matet-Mathias forcing restricted to a Matet-adequate family.

math.LO↗

arXiv2024

Initial Tukey structure below a stable ordered-union ultrafilter

Answering a question of Dobrinen and Todorcevic, we prove that below any stable ordered-union ultrafilter $\mathcal{U}$, there are exactly four nonprincipal Tukey classes: $[\mathcal{U}], [\mathcal{U}_{\operatorname{min}}], [\mathcal{U}_{\operatorname{max}}]$, and $[\mathcal{U}_{\operatorname{minmax}}]$. This parallels the classification of ultrafilters Rudin-Keisler below $\mathcal{U}$ by Blass. A key step in the proof involves modifying the proof of a canonization theorem of Klein and Spinas for Borel functions on $\mathrm{FIN}^{[\infty]}$ to obtain a simplified canonization theorem for fronts on $\mathrm{FIN}^{[\infty]}$, recovering Lefmann's canonization for fronts of finite uniformity rank as a special case. We use this to classify the Rudin-Keisler classes of all ultrafilters Tukey below $\mathcal{U}$, which is then applied to achieve the main result.

math.LO↗