Hušek defines a space X to have a small diagonal if each uncountable subset of X² disjoint from the diagonal has an uncountable subset whose closure is disjoint from the diagonal. Hušek proved that a compact space of weight ω₁ which has a small diagonal will be metrizable, but it remains an open problem to determine if the weight restriction is necessary. It has been shown to be consistent that each compact space with a small diagonal is metrizable; in particular, Juhász and Szentmiklóssy proved...

We establish two new Easton theorems for the least supercompact cardinal that are consistent with the level by level equivalence between strong compactness and supercompactness. These theorems generalize Theorem 1 in our earlier paper [Math. Logic Quart. 51 (2005)]. In both our ground model and the model witnessing the conclusions of our present theorems, there are no restrictions on the structure of the class of supercompact cardinals.

Given an ideal $ℐ$ on $\omega$ let $𝔞\left(ℐ\right)$ ($\overline{𝔞}\left(ℐ\right)$) be minimum of the cardinalities of infinite (uncountable) maximal $ℐ$-almost disjoint subsets of ${\left[\omega \right]}^{\omega }$. We show that $𝔞\left({ℐ}_h\right)>\omega$ if ${ℐ}_h$ is a summable ideal; but $𝔞\left({𝒵}_{\overrightarrow{\mu }}\right)=\omega$ for any tall density ideal ${𝒵}_{\overrightarrow{\mu }}$ including the density zero ideal $𝒵$. On the other hand, you have $𝔟\le \overline{𝔞}\left(ℐ\right)$ for any analytic $P$-ideal $ℐ$, and $\overline{𝔞}\left({𝒵}_{\overrightarrow{\mu }}\right)\le 𝔞$ for each density ideal ${𝒵}_{\overrightarrow{\mu }}$. For each ideal $ℐ$ on $\omega$ denote ${𝔟}_{ℐ}$ and ${𝔡}_{ℐ}$ the unbounding and dominating numbers of $\langle {\omega }^{\omega },{\le }_{ℐ}\rangle$ where $f{\le }_{ℐ}g$ iff $\{n\in \omega :f(n)>g(n\left)\right\}\in ℐ$. We show that ${𝔟}_{ℐ}=𝔟$ and ${𝔡}_{ℐ}=𝔡$ for each analytic $P$-ideal $ℐ$. Given a Borel ideal $ℐ$ on...

This paper is a continuation of [19], where the divisibility criteria for initial prime numbers based on their representation in the decimal system were formalized. In the current paper we consider all primes up to 101 to demonstrate the method presented in [7].

Continuing the earlier research [Fund. Math. 129 (1988) and 149 (1996)] we give some information about extending automorphisms of models of PA to cofinal extensions.

By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, $EF{G}_{\omega ₁}(,)$, is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and $EF{G}_{\omega ₁}(,)$ is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if $2^{\aleph ₀}<2^{\aleph ₃}$, T is a countable complete...

A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as $X=A{\bowtie }_{x}B$ where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique fixed point...

We consider the question of when $X_M=X$, where $X_M$ is the elementary submodel topology on X ∩ M, especially in the case when $X_M$ is compact.

We prove in ZFC that every $ℳ\cap 𝒩$ additive set is $𝒩$ additive, thus we solve Problem 20 from paper [Weiss T., A note on the intersection ideal $ℳ\cap 𝒩$, Comment. Math. Univ. Carolin. 54 (2013), no. 3, 437-445] in the negative.

We study polychromatic Ramsey theory with a focus on colourings of [ω 2]2. We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2ω = ω 2 and ${\omega }_2{\to }^{poly}{\left(\alpha \right)}_{{\aleph }_0-bdd}^2$ for every α <ω 2; (2) 2ω = ω 2 and ${\omega }_2{\nrightarrow }^{poly}{\left({\omega }_1\right)}_{2-bdd}^2$ .

We introduce a very weak version of the square principle which may hold even under failure of the generalized continuum hypothesis. Under this weak square principle, we give a new characterization (Theorem 10) of partial orderings with κ-Freese-Nation property (see below for the definition). The characterization is not a ZFC theorem: assuming Chang’s Conjecture for ${\aleph }_{\omega }$, we can find a counter-example to the characterization (Theorem 12). We then show that, in the model obtained by adding Cohen reals,...

Morley’s trisector theorem states that “The points of intersection of the adjacent trisectors of the angles of any triangle are the vertices of an equilateral triangle” [10]. There are many proofs of Morley’s trisector theorem [12, 16, 9, 13, 8, 20, 3, 18]. We follow the proof given by A. Letac in [15].