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

Piece selection and cardinal arithmetic