Open covers and the square bracket partition relation

Abstract:

An open cover $\mathcal {U}$ of an infinite separable metric space $X$ is an $\omega$-cover of $X$ if $X\not \in \mathcal {U}$ and for every finite subset $F$ of $X$ there is a $U\in \mathcal {U}$ such that $F\subseteq U$. Let $\Omega$ be the collection of $\omega$–covers of $X$. We show that the partition relation $\Omega \rightarrow [\Omega ]^2_2$ holds if, and only if, the partition relation $\Omega \rightarrow [\Omega ]^2_3$ holds.