Related papers: Stable ordered union ultrafilters and $\mathrm{cov…

It is shown that there exists a complete, atomless, sigma-centered Boolean algebra, which does not contain any regular countable subalgebra if and only if there exist a nowhere dense ultrafilter. Therefore the existence of such algebras is…

We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbf{\Delta}^1_2$ maximal independent set. As a main…

An earlier paper, entitled "P-hierarchy on $\beta\omega$", investigated the relations between ordinal ultrafilters and the so-called P-hierarchy. This study is continued in the present paper and focuses on the aspects of characterization of…

It is proved that the existence of a countable extremally disconnected Boolean topological group containing a family of open subgroups whose intersection has empty interior implies the existence of a rapid ultrafilter.

The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa. Then for every regular lambda < pp^+_{Gamma(kappa)} (mu) there is an increasing…

We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V is a model…

We study ultrafilters on countable sets and reaping families which are indestructible by Sacks forcing. We deal with the combinatorial characterization of such families and we prove that every reaping family of size smaller than the…

Let $G$ be an abelian group, and $F$ a downward directed family of subsets of $G$. The finest topology $\mathcal{T}$ on $G$ under which $F$ converges to $0$ has been described by I.Protasov and E.Zelenyuk. In particular, their description…

We investigate the question of whether $\mathbb Q$ carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this…

In this paper, we provide a combinatorial characterization of the elements of Schur ultrafilters on countable commutative groups. Using this characterization, we construct a free Schur ultrafilter on $\mathbb Z$ that is not infinitary…

Theorem: There is a {\em complete sentence} $\phi$ of $L_{\omega_1,\omega}$ such that $\phi$ has maximal models in a set of cardinals $\lambda$ that is cofinal in the first measurable $\mu$ while $\phi$ has no maximal models in any $\chi…

An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to…

Let $p$ be a prime number. A saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is said to be supersolvable if there is a series $1 = S_0 \le S_1 \le \dots \le S_m = S$ of subgroups of $S$ such that $S_i$ is strongly…

A unified theory is outlined for the cuprates, Fe-based, and related superconductors. Their low-energy excitations are approached in terms of auxiliary particles representing combinations of atomic-like electron configurations, and the…

We discuss the connection between various orders on the class of all the ultrafilters and certain compactness properties of abstract logics and of topological spaces. We present a model theoretical characterization of Comfort order. We…

The tower number $\mathfrak t$ and the ultrafilter number $\mathfrak u$ are cardinal characteristics from set theory. They are based on combinatorial properties of classes of subsets of~$\omega$ and the almost inclusion relation…

Using a new concept of conglomerated filter we demonstrate in a purely combinatorial way that none of Erd\"{o}s-Ulam filters or summable filters can be generated by a single statistical measure and consequently they cannot be represented as…

We consider, for infinite cardinals kappa and alpha <= kappa^+, the group Pi(kappa,< alpha) of sequences of integers, of length kappa, with non-zero entries in fewer than alpha positions. Our main result tells when Pi(kappa,< alpha) can be…

In this note we consider an arbitrary families of sets of $s_0$ ideal introduced by Marczewski-Szpilrajn. We show that in any uncountable Polish space $X$ and under some combinatorial and set theoretical assumptions ($cov(s_0)=\c$ for…

We study the structure of the Rudin-Frolik order on countably complete ultrafilters under the assumption that this order is directed. This assumption, called the Ultrapower Axiom, holds in all known canonical inner models. It turns out that…