Piece selection and cardinal arithmetic

We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions Abe and Usuba. In particular, show that if λ ≥ 2 κ $\lambda \ge 2^\kappa$ , then (a) I $I_{\kappa \lambda }$ is not (λ, 2)-distributive, (b) + → ( ) ω }^+ \rightarrow (I_{\kappa }^+)^2_\omega$ does hold.