The additivity spectrum $ADD\left(ℐ\right)$ of an ideal $ℐ\subset 𝒫\left(I\right)$ is the set of all regular cardinals $\kappa$ such that there is an increasing chain $\{A_{\alpha }:\alpha <\kappa \}\subset ℐ$ with ${\bigcup }_{\alpha <\kappa }{A}_{\alpha }\notin ℐ$. We investigate which set $A$ of regular cardinals can be the additivity spectrum of certain ideals. Assume that $ℐ=ℬ$ or $ℐ=𝒩$, where $ℬ$ denotes the $\sigma$-ideal generated by the compact subsets of the Baire space ${\omega }^{\omega }$, and $𝒩$ is the ideal of the null sets. We show that if $A$ is a non-empty progressive set of uncountable regular cardinals and $pcf\left(A\right)=A$, then $ADD\left(ℐ\right)=A$ in some c.c.c generic extension...

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...

Let $(X,𝕀)$ be a Polish ideal space and let $T$ be any set. We show that under some conditions on a relation $R\subseteq T^2\times X$ it is possible to find a set $A\subseteq T$ such that $R\left(A^2\right)$ is completely $𝕀$-nonmeasurable, i.e, it is $𝕀$-nonmeasurable in every positive Borel set. We also obtain such a set $A\subseteq T$ simultaneously for continuum many relations ${\left(R_{\alpha }\right)}_{\alpha <2^{\omega }}.$ Our results generalize those from the papers of K. Ciesielski, H. Fejzić, C. Freiling and M. Kysiak.