Motivated by an application to the unconditional basic sequence problem appearing in our previous paper, we introduce analogues of the Laver ideal on ℵ₂ living on index sets of the form ${\left[{\aleph }_k\right]}^{\omega }$ and use this to refine the well-known high-dimensional polarized partition relation for ${\aleph }_{\omega }$ of Shelah.

Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any $f:{\cup }_{n<\omega }{\left[X\right]}_{\subset }^n\to \gamma$ with $X\subset P_{\kappa }\lambda$ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with $|f^{\text{'}\text{'}}{\left[Y\right]}_{\subset }^n|=1$ for any n < ω.

A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.

We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a $T_3$ Lindelöf topology can be partitioned into two bases while there exists a consistent example of a first-countable, 0-dimensional, Hausdorff space of size $2^{\omega }$ and weight ${\omega }_1$ which admits a point countable base without a partition to two bases.

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.

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...

If G is a group then the abelian subgroup spectrum of G is defined to be the set of all κ such that there is a maximal abelian subgroup of G of size κ. The cardinal invariant A(G) is defined to be the least uncountable cardinal in the abelian subgroup spectrum of G. The value of A(G) is examined for various groups G which are quotients of certain permutation groups on the integers. An important special case, to which much of the paper is devoted, is the quotient of the full symmetric group by the...

We study a notion of potential isomorphism, where two structures are said to be potentially isomorphic if they are isomorphic in some generic extension that preserves stationary sets and does not add new sets of cardinality less than the cardinality of the models. We introduce the notion of weakly semi-proper trees, and note that there is a strong connection between the existence of potentially isomorphic models for a given complete theory and the existence of weakly semi-proper trees. ...

Module is said to be small if it is not a union of strictly increasing infinite countable chain of submodules. We show that the class of all small modules over self-injective purely infinite ring is closed under direct products whenever there exists no strongly inaccessible cardinal.

We continue our work on weak diamonds [J. Appl. Anal. 15 (1009)]. We show that $2^{\omega }=\aleph ₂$ together with the weak diamond for covering by thin trees, the weak diamond for covering by meagre sets, the weak diamond for covering by null sets, and “all Aronszajn trees are special” is consistent relative to ZFC. We iterate alternately forcings specialising Aronszajn trees without adding reals (the NNR forcing from [“Proper and Improper Forcing”, Ch. V]) and < ω₁-proper ${}^{\omega }\omega$-bounding forcings adding reals. We show...

Recall that a P-set is a closed set X such that the intersection of countably many neighborhoods of X is again a neighborhood of X. We show that if 𝔱 = 𝔠 then there is a minimal right ideal of (βℕ,+) that is also a P-set. We also show that the existence of such P-sets implies the existence of P-points; in particular, it is consistent with ZFC that no minimal right ideal is a P-set. As an application of these results, we prove that it is both consistent with and independent of ZFC that the shift...