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)

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.

We study a normal ideal on Pκ that is defined in terms of games.