Maximal eventually different families for uniformly weak Ramsey ideals

TL;DR

This paper investigates the existence of maximal eventually different families of functions under certain ideals on natural numbers, introducing a class of ideals called uniformly weak Ramsey ideals and establishing conditions for such families to exist.

Contribution

The paper introduces uniformly weak Ramsey ideals and proves the existence of closed maximal eventually different families for these ideals, including all $F_\sigma$ ideals and certain Fubini products.

Findings

Existence of closed $ ext{I}$-maximal eventually different families for uniformly weak Ramsey ideals.

Such families exist for all $F_\sigma$ ideals.

They also exist for Fubini products $ ext{fin}^eta$ with $eta< ext{omega}_1$.

Abstract

We study $\mathcal I$-maximal eventually different families of functions from the set of natural numbers into itself where $\mathcal I$ is an arbitrary ideal on the set of natural numbers that includes the ideal of all finite sets $\mathrm{fin}$. We introduce the class of uniformly weak Ramsey ideals and prove that there exists a closed $\mathcal I$-maximal eventually different family if $\mathcal I$ belongs to this class; this is the case for arbitrary $F_\sigma$ ideals and Fubini products $\mathrm{fin}^{\alpha}$ with $\alpha<\omega_1$.