# Piece selection and cardinal arithmetic

## TL;DR

This paper explores how piece selection principles influence cardinal arithmetic, providing new results on distributivity and partition properties of certain ideals related to Shelah's work.

## Contribution

It introduces novel connections between piece selection principles and cardinal arithmetic, addressing open questions by Abe and Usuba.

## Key findings

- If λ ≥ 2^κ, then I_{κ, λ} is not (λ, 2)-distributive.
- Under the same condition, I_{κ, λ}^+ does not satisfy a certain partition property.
- The results extend understanding of the interplay between combinatorial principles and cardinal characteristics.

## Abstract

We study the effects of piece selection principles on cardinal arithmetic (Shelah style). As an application, we discuss questions of Abe and Usuba. In particular, 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.