Related papers: Club guessing for dummies

We prove a weakened version of the reflection of Reinhardt cardinals by super Reinhardt cardinals: Let $M=(V^M,P)$ be a countable model of second order set theory $\mathsf{ZF}_2$ (with universe $V^M$ and classes $P$) which models "$\kappa$…

We conjecture that every infinite group $G$ can be partitioned into countably many cells $G=\bigcup_{n\in\omega}A_n$ such that $cov(A_nA_n^{-1})=|G|$ for each $n\in\omega$. Here $cov(A)=\min\{|X|:X\subseteq G, G=XA\}$. We confirm this…

Let $\chi$ be a real primitive character to the modulus $D$. It is proved that $$ L(1,\chi)\gg (\log D)^{-2022} $$ where the implied constant is absolute and effectively computable. In the proof, the lower bound for $L(1,\chi)$ is first…

Let $\kappa$,$\lambda$ be regular uncountable cardinals such that $\lambda > \kappa^+$ is not a successor of a singular cardinal of low cofinality. We construct a generic extension with $s(\kappa) = \lambda$ starting from a ground model in…

The main purpose of the paper is the proof of a cardinal inequality for a space with points $G_\delta$, obtained with the help of a long version of the Menger game. This result improves a similar one of Scheepers and Tall.

We discuss how singular can cardinals be in absence of the axiom of choice. We show that, contrasting with known negative consistency results (of Gitik and others), certain positive results are provable. Then we pose some problems.

Characterizations of skeletally Dugundji spaces and Dugundji spaces are given in terms of club collections, consisting of countable families of co-zero sets. For example, a Tychonoff space $X$ is skeletally Dugundji if and only if there…

We prove Beurling's theorem for the full group $SL(2,\R)$. This is the {\em master theorem} in the quantitative uncertainty principle as all the other theorems of this genre follow from it.

The paper extends Birkhoff's theorem on doubly stochastic matrices to some countable families of discrete probability spaces with nonempty intersections. We join every two elements lying in the same probability space by an edge and…

We prove that the perfect set dichotomy theorem holds in the Solovay model $V ((\omega^\omega)^{V[G]})$. Namely, for every equivalence relation $E$ on $\mathbb{R}$, either $\mathbb{R}/E$ is well-orderable or there exists a perfect set…

A club consisting of former regulars is added to an inaccessible cardinal, without changing cofinalities outside it. The initial assumption is optimal. A variation of the Radin forcing without a top measurable cardinal is introduced for…

We consider families F of sequences converging to +infinity that F satisfies the following condition (C): (C): if an open set U in the real line is unbounded above then there exists a sequence belonging to F, which has an infinite number of…

The Skolem Problem asks, given an integer linear recurrence sequence (LRS), to determine whether the sequence contains a zero term or not. Its decidability is a longstanding open problem in theoretical computer science and automata theory.…

We study the geometry of unitary Shimura varieties without assuming the existence of an ordinary locus. We prove, by a simple argument, the existence of canonical subgroups on a strict neighborhood of the $\mu$-ordinary locus (with an…

We prove an extension of the Moore-Schmidt theorem on the triviality of the first cohomology class of cocycles for the action of an arbitrary discrete group on an arbitrary measure space and for cocycles with values in an arbitrary compact…

Let $\mu$ be a strong limit singular cardinal. We prove that if $2^{\mu} > \mu^+$ then $\binom{\mu^+}{\mu}\to \binom{\tau}{\mu}_{<{\rm cf}(\mu)}$ for every ordinal $\tau<\mu^+$. We obtain an optimal positive relation under $2^\mu = \mu^+$,…

We generalize the Hart-Shelah example \cite{HaSh:323} to higher infinitary logics. We build, for each natural number $k\geq 2$ and for each infinite cardinal $\lambda$, a sentence $\psi_k^\lambda$ of the logic $L_{(2^\lambda)^+,\omega}$…

We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not…

In this paper we continue the study in [Gilton-Levine-Stejskalova] of compactness and incompactness principles at double successors, focusing here on the case of double successors of singulars of countable cofinality. We obtain models which…

We investigate an extension of ZFC set theory (in an extended language) that stipulates the existence of a proper class of indiscernibles over the universe. One of the main results of the paper shows that the purely set-theoretical…