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 $\alpha<\omega_1$.
Categories: math.LO