Related papers: Characterizations of the weakly compact ideal on $…

We extend to singular cardinals the model-theoretical relation $\lambda \stackrel{\kappa}{\Rightarrow} \mu$ introduced in P. Lipparini, The compactness spectrum of abstract logics, large cardinals and combinatorial principles, Boll. Unione…

Let M denote the ideal of first category subsets of R. We prove that min{card X: X \subseteq R, X \not\in M} is the smallest cardinality of a family S \subseteq {0,1}^\omega with the property that for each f: \omega -> \bigcup_{n \in…

For each countable ordinal $\alpha \ge 2$, the ideals $\mathsf{conv}_\alpha$ were introduced in ``Critical ideals for countable compact spaces'' (to appear in Fund. Math., see also arXiv:2503.12571) to characterize compact countable spaces…

We provide analogues of the results from [FMR11, CMMR13] in the reference list (which correspond to the case $\kappa = \omega$) for arbitrary $\kappa$-Souslin quasi-orders on any Polish space, for $\kappa$ an infinite cardinal smaller than…

This work is a part of my upcoming thesis [7]. We establish an equiconsistency between (1) weak indestructibility for all $\kappa +2$-degrees of strength for cardinals $\kappa $ in the presence of a proper class of strong cardinals, and (2)…

We give a development of the fine structure of mice with long extenders, to the level of $\kappa^+$-supercompact cardinals $\kappa$. We do this using a hierarchy with features more analogous to those familiar in the short extender context…

We study Structural Reflection beyond Vop\v{e}nka's Principle, at the level of almost-huge cardinals and higher, up to rank-into-rank embeddings. We identify and classify new large cardinal notions in that region that correspond to some…

Generalizing Keisler's notion of regularity for ultrafilters, Taylor introduced degrees of regularity for ideals and showed that a countably complete nonregular ideal on $\omega_1$ must be somewhere $\omega_1$-dense. We prove a dichotomy…

For a discrete group $G$, we use the natural correspondence between ideals in the Boolean algebra $ \mathcal{P}_G$ of subsets of $G$ and closed subsets in the Stone-$\check{C}$ech compactifi-cation $\beta G$ as a right topological semigroup…

In this paper, we characterize the possible cofinalities of the least $\lambda$-strongly compact cardinal. We show that, on the one hand, for any regular cardinal, $\delta$, that carries a $\lambda$-complete uniform ultrafilter, it is…

This paper is devoted to studies of IwQN-spaces and some of their cardinal characteristics. Recently, \v{S}upina proved that I is not a weak P-ideal if and only if any topological space is an IQN-space. Moreover, under…

We extend prior results of Cody-Eskew, showing the consistency of GCH with the statement that for all regular cardinals $\kappa \leq \lambda$, where $\kappa$ is the successor of a regular cardinal, there is a rigid saturated ideal on…

We investigate forms of filter extension properties in the two-cardinal setting involving filters on $P_\kappa(\lambda)$. We generalize the filter games introduced by Holy and Schlicht in \cite{HolySchlicht:HierarchyRamseyLikeCardinals} to…

We provide a model where u(\kappa) < 2^{\kappa} for a supercompact cardinal \kappa. Garti and Shelah have provided a sketch of how to obtain such a model by modifying the construction in a paper of Dzamonja and Shelah; we provide here a…

We characterize exactly the compactness properties of the product of \kappa\ copies of the space \omega\ with the discrete topology. The characterization involves uniform ultrafilters, infinitary languages, and the existence of nonstandard…

We prove that superclub implies $\mathfrak{s}=\aleph_1$. More generally, superclub at a successor of a weakly compact cardinal implies $\mathfrak{s}_\kappa=\kappa^+$. Based on this statement, we separate tiltan from superclub at a successor…

It is known that there are many notions of largeness in a semigroup that own rich combinatorial properties. In this paper, we focus on partition and almost disjoint properties of these notions. One of the most remarkable results with…

Answering questions raised in \cite{Leonetti, Uzcategui} we characterize ideals $\mathcal I\subseteq \mathcal P(\omega)$ such that $c_{0,\mathcal I}$ is complemented in $\ell_\infty$ as exactly those ideals for which the space $K_{\mathcal…

Following \cite{bagaria2019large}, given cardinals $\kappa<\lambda$, we say $\kappa$ is a club $\lambda$-Berkeley cardinal if for every transitive set $N$ of size $<\lambda$ such that $\kappa\subseteq N$, there is a club $C\subseteq \kappa$…

We show that in the $\mathbb{P}_{\max}$ extension of a certain Chang-type model of determinacy, if $\kappa\in\{\omega_1, \omega_2, \omega_3\}$, then the restriction of the club filter on $\kappa\cap\mathrm{Cof}(\omega)$ to HOD is an…