Three sets occurring in functional analysis are shown to be of class PCA (also called ${\Sigma }_2^1$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].

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 of the...

We study several perfect set properties of the Baire space which follow from the Ramsey property $\omega \to {\left(\omega \right)}^{\omega }$. In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

We study splitting, infinitely often equal (ioe) and refining families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set theorems. We succeed for $G_{\delta }$ hereditary splitting families and for analytic countably ioe families. We construct several examples of small closed ioe and refining families.

Under Martin’s axiom, collapsing of the continuum by Sacks forcing $𝕊$ is characterized by the additivity of Marczewski’s ideal (see [4]). We show that the same characterization holds true if $𝔡=𝔠$ proving that under this hypothesis there are no small uncountable maximal antichains in $𝕊$. We also construct a partition of ${}^{\omega }2$ into $𝔠$ perfect sets which is a maximal antichain in $𝕊$ and show that $s^0$-sets are exactly (subsets of) selectors of maximal antichains of perfect sets.

We investigate properties of permitted trigonometric thin sets and construct uncountable permitted sets under some set-theoretical assumptions.

By an ${\omega }_1$- tree we mean a tree of power ${\omega }_1$ and height ${\omega }_1$. Under CH and $2^{{\omega }_1}>{\omega }_2$ we call an ${\omega }_1$-tree a Jech-Kunen tree if it has κ-many branches for some κ strictly between ${\omega }_1$ and $2^{{\omega }_1}$. In this paper we prove that, assuming the existence of one inaccessible cardinal, (1) it is consistent with CH plus $2^{{\omega }_1}>{\omega }_2$ that there exist Kurepa trees and there are no Jech-Kunen trees, which answers a question of [Ji2], (2) it is consistent with CH plus $2^{{\omega }_1}={\omega }_4$ that there only exist Kurepa trees with ${\omega }_3$-many branches, which answers another...

Given a complete, superstable theory, we distinguish a class P of regular types, typically closed under automorphisms of ℭ and non-orthogonality. We define the notion of P-NDOP, which is a weakening of NDOP. For superstable theories with P-NDOP, we prove the existence of P-decompositions and derive an analog of the first author's result in Israel J. Math. 140 (2004). In this context, we also find a sufficient condition on P-decompositions that implies non-isomorphic models. For this, we investigate...