Non‐saturation of the non‐stationary ideal on P κ (λ) with λ of countable cofinality

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 P κ (λ) to the set of all a with \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{cf}(\sup (a\cap \kappa)) = \omega$\end{document} is not λ ++ ‐saturated (and even not \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$2^{{(\lambda ^{<\kappa }})}$\end{document} ‐saturated in case 2 λ = λ + ). We actually prove the stronger result that there is \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$Q\subseteq \mathrm{NG}_{\kappa ,\lambda }^+$\end{document} with | Q | = λ ++ such that A ∩ B is a non‐cofinal subset of P κ (λ) for any two distinct members A , B of Q , where NG κ, λ denotes the game ideal on P κ (λ). We also remark that for κ > ω 1 , adding λ +3 Cohen subsets of ω 1 to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {L}$\end{document} makes NG κ, λ λ +3 ‐saturated.