A model in which strongness ofκ is indestructible under κ+ -weakly closed forcing notions satisfying the Prikry condition is constructed. This is applied to solve a question of Hajnal on the number of elements of {λ δ |2 δ <λ}.

Since the work of Gödel and Cohen, which showed that Hilbert's First Problem was independent of the usual assumptions of mathematics, there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond and square discovered by Jensen. Simultaneously, attempts have been made to find suitable (...) natural strengthenings of ZFC, primarily by Large Cardinal or Reflection Axioms. These two directions have tension between them in that Jensen's principles, which tend to suggest a rather rigid mathematical universe, are at odds with reflection properties. A third development was the discovery by Shelah of "PCF Theory", a generalization of cardinal arithmetic that is largely determined inside ZFC. In this paper we consider interactions between these three theories in the context of singular cardinals, focusing on the various implications between square and scales, and on consistency results between relatively strong forms of square and stationary set reflection. (shrink)

Here we deal with some problems posed by Matet. The first section deals with the existence of stationary subsets of [λ]<κ with no unbounded subsets which are not stationary, where, of course, κ is regular uncountable ≤λ. In the second section we deal with the existence of such clubs. The proofs are easy but the result seems to be very surprising. Theorem 1.2 was proved some time ago by Baumgartner (see Theorem 2.3 of [Jo88]) and is presented here for the (...) sake of completeness. (shrink)

We use ideas of Fred Galvin to show that under Martin's axiom, there is a prime ideal on Pω (λ) with the partition property for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}\end{document}.

Given a regular infinite cardinal κ and a cardinal λ > κ, we study fine ideals H on Pκ that satisfy the square brackets partition relation equation image, where μ is a cardinal ≥2.