相关论文: Decreasing chains without lower bounds in Rudin-Fr…

The aim of this paper is to construct chains of length $(2^\kappa)^+$ in the sense of Rudin-Frol\'ik order in $\beta \kappa$ for $\kappa$ regular.

We show that there is a set of $2^{2^{\kappa}}$ ultrafilters incomparable in Rudin-Frol\'ik order of $\beta \kappa \setminus \kappa$, where $\kappa$ is regular, for which no subset with more than one element has an infimum.

We point out that a construction by Butkovi\v{c}ov\'a of a chain of length $\mathfrak{c}^+$ in the Rudin-Frol\'ik order on $\beta\omega$ can easily be adapted to produce, given an uncountable cardinal $\kappa$, a chain of length…

The aim of this paper is to construct ultrafilters without immediate predecessors in the Rudin-Frolik order in $\beta \kappa\setminus \kappa$, where $\kappa$ is a regular cardinal. This generalizes the problem posed by Peter Simon more than…

Assuming an instance of the Brodsky-Rinot proxy principle holding at a regular uncountable cardinal $\kappa$, we construct $2^\kappa$-many pairwise non-embeddable minimal non-$\sigma$-scattered linear orders of size $\kappa$. In particular,…

We use the core model for sequences of measures to prove a new lower bound for the consistency strength of the failure of the SCH: THEOREM (i) If there is a singular strong limit cardinal $\kappa$ such that $2^\kappa > kappa^+$ then there…

We obtain lower bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta function for all k > 1. Previously such lower bounds were known only for rational values of k, with the bounds depending on the height of the…

The main result of this paper is to show that for any compact space $X$ and any family $\{f_\alpha \colon \alpha< 2^\kappa, f_\alpha \colon X \stackrel{onto}{\longrightarrow} \beta \kappa\}$ of open mappings there exists a point $x \in X$…

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders…

Suppose $\kappa$ is a regular cardinal and $\bar a=\langle \mu_i: i<\kappa \rangle$ is a non-decreasing sequence of regular cardinals. We study the set of possible cofinalities of cuts Pcut$(\bar a)=\{(\lambda_1, \lambda_2):$ for some…

A simplified construction is presented for Komj\'ath's result that for every uncountable cardinal $\kappa$, there are $2^\kappa$ graphs of size $\kappa$ none of them being a minor of another.

Shelah has shown that there are no chains of length $\omega_3$ increasing modulo finite in ${}^{\omega_2}\omega_2$. We improve this result to sets. That is, we show that there are no chains of length $\omega_3$ in $[\omega_2]^{\aleph_2}$…

We show that for any union-closed family $\mathcal{F} \subseteq 2^{[n]}, \mathcal{F} \neq \{\emptyset\}$, there exists an $i \in [n]$ which is contained in a $0.01$ fraction of the sets in $\mathcal{F}$. This is the first known constant…

We show that $\mathrm{MA}_{\kappa}$ implies that each collection of ${P}_{\mathfrak c}$-points of size at most $\kappa$ which has a $P_{\mathfrak c}$-point as an $RK$ upper bound also has a ${P}_{\mathfrak c}$-point as an $RK$ lower bound.

It is proved that the Continuum Hypothesis implies that any sequence of rapid P-points of length $<{\mathfrak c}^{+}$ which is increasing with respect to the Rudin-Keisler ordering is bounded above by a rapid P-point. This is an improvement…

Let $\kappa$ be an uncountable cardinal such that $2^{<\kappa} = \kappa$ or just ${\rm cf}(\kappa) > \omega$, $2^{2^{<\kappa}}= 2^\kappa$, and $([\kappa]^\kappa, \supseteq)$ collapses $2^\kappa$ to $\omega$. We show under these assumptions…

We study the generalized dominating number $\mathfrak{d}_{\mu}$ at a singular cardinal $\mu$ of cofinality $\kappa$. We show two lower bounds: in ZFC, $\mathrm{cf}([\mu]^\kappa,\subseteq) \leq \mathfrak{d}_{\mu}$, and under mild…

We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all…

We show it is consistent with $\ZFC$ that there is an everywhere Kurepa line which is order isomorphic to all of its dense $\aleph_2$-dense suborders. Moreover, this Kurepa line does not contain any Aronszajn suborder. We also show it is…

In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an…