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...

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 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,...

We study a higher-dimensional version of the standard notion of a gap formed by a finite sequence of ideals of the quotient algebra 𝓟(ω)/fin. We examine different types of such objects found in 𝓟(ω)/fin both from the combinatorial and the descriptive set-theoretic side.

Three complete characteristics of couples of nonadditive cuts such that $\underline{J\times K}\ne \overline{JtimesK}$ are given. The equality $\overline{J\times K}=J!K$ is proved for all couples of nonadditive cuts. Some examples of nonadditive cuts are described.