Let κ be a regular uncountable cardinal and λ be a cardinal greater than κ. We show that if 2 <κ ≤ M(κ, λ), then ◇ κ,λ holds, where M(κ, λ) equals $\lambda ^{\aleph }0$ if cf(λ) ≥ κ, and $(\lambda ^{+})^{\aleph _{0}}$ otherwise.

Given a regular cardinal κ > ω 1 and a cardinal λ with κ ≤ cf (λ) < λ, we show that NS κ,λ | T is not λ+-saturated, where T is the set of all ${a\in P_\kappa (\lambda)}$ such that ${| a | = | a \cap \kappa|}$ and ${{\rm cf} \big( {\rm sup} (a\cap\kappa)\big) = {\rm cf} \big({\rm sup} (a)\big) = \omega}$.

Given a regular uncountable cardinal κ and a cardinal λ > κ of cofinality ω, we show that the restriction of the non-stationary ideal on Pκ to the set of all a with equation image is not λ++-saturated . We actually prove the stronger result that there is equation image with |Q| = λ++ such that A∩B is a non-cofinal subset of Pκ for any two distinct members A, B of Q, where NGκ, λ denotes the game ideal on Pκ. (...) We also remark that for κ > ω1, adding λ+3 Cohen subsets of ω1 to equation image makes NGκ, λ λ+3-saturated. (shrink)

We study normal ideals on Pκ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${P_{\kappa} }$$\end{document} that are defined in terms of games of uncountable length.

Let $\kappa $ be a regular uncountable cardinal, and a cardinal greater than or equal to $\kappa $. Revisiting a celebrated result of Shelah, we show that if is close to $\kappa $ and (= the least size of a cofinal subset of ) is greater than, then can be represented (in the sense of pcf theory) as a pseudopower. This can be used to obtain optimal results concerning the splitting problem. For example we show that if and, then no (...) $\kappa $ -complete ideal on is weakly -saturated. (shrink)