Related papers: How far is almost strong compactness from strong c…
We give the definition of Woodin for strong compactness cardinals, the Woodinised version of strong compactness, and we prove an analogue of Magidor's identity crisis theorem for the first strongly compact cardinal.
We construct a normal countably tight $T_1$ space $X$ with $t(X_\delta) >2^\omega$. This is an answer to the question posed by Dow-Juh\'asz-Soukup-Szentmikl\'ossy-Weiss. We also show that if the continuum is not so large, then the tightness…
We prove several results giving lower bounds for the large cardinal strength of a failure of the singular cardinal hypothesis. The main result is the following theorem: Theorem: Suppose $\kappa$ is a singular strong limit cardinal and…
We strengthen a result of Bagaria and Magidor~\cite{MR3152715} about the relationship between large cardinals and torsion classes of abelian groups, and prove that (1) the \emph{Maximum Deconstructibility} principle introduced in…
We give a partial solution to a question by Alas, Junqueria and Wilson by proving that under PFA the one-point compactification of a locally compact, discretely generated and countably tight space is also discretely generated. After this,…
Motivated by the minimal tower problem, an earlier work studied diagonalizations of covers where the covers are related to linear quasiorders (tau-covers). We deal with two types of combinatorial questions which arise from this study. 1.…
We show that unitary groups of II$_1$ factors and of properly infinite von Neumann algebras have strong uncountable cofinality. In particular, we obtain a short alternative proof for the strong uncountable cofinality of…
A countable, bounded degree graph is almost finite if it has a tiling with isomorphic copies of finitely many F\o lner sets, and we call it strongly almost finite, if the tiling can be randomized so that the probability that a vertex is on…
Relative to class many supercompact cardinals, we construct a model of $\ZFC+\GCH$ where for every singular cardinal $\delta$ of countable cofinality and every regular uncountable $\mu<\delta$ there are stationarily many non-approachable…
If $\kappa$ is regular and $2^{<\kappa}\leq\kappa^+$, then the existence of a weakly presaturated ideal on $\kappa^+$ implies $\square^*_\kappa$. This partially answers a question of Foreman and Magidor about the approachability ideal on…
Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $L(\mathbb R)$ is absolute for proper forcing. Here, we study the…
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…
We have defined almost separable space. We show that like separability, almost separability is $c$ productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts…
The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa$ is…
We produce a model where every supercompact cardinal is $C^{(1)}$-supercompact with inaccessible targets. This is a significant improvement of the main identity-crises configuration obtained in \cite{HMP} and provides a definitive answer to…
We prove that the strong polarized relation of $\theta$ above $\omega$ applied simultaneously for every cardinal in the interval $[\aleph_1,\aleph]$ is consistent. We conclude that this positive relation is consistent for every cardinal…
This is the second combinatorial proof of the compactness theorem for singular from 1977. In fact it gives a somewhat stronger theorem.
We solve a long standing question due to Arhangel'skii by constructing a compact space which has a $G_\delta$ cover with no continuum-sized ($G_\delta$)-dense subcollection. We also prove that in a countably compact weakly Lindel\"of normal…
We prove upper bounds for the spread, the Lindel\"of number and the weak Lindel\"of number of the $G_\delta$-topology on a topological space and apply a few of our bounds to give a short proof to a recent result of Juh\'asz and van Mill…
We determine the large cardinal consistency strength of the existence of a $\lambda$-supercompact cardinal $\kappa$ such that GCH fails at $\lambda$. Indeed, we show that the existence of a $\lambda$-supercompact cardinal $\kappa$ such that…