Open covers and the square bracket partition relation

An open cover ${\cal U}$ of an infinite separable metric space $X$ is an $\omega$-cover of $X$ if $X\not \in{\cal U}$ and for every finite subset $F$ of $X$ there is a $U\in {\cal U}$ 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.