Open covers and the square bracket partition relation

An open cover u of an infinite separable metric space X is an w-cover of X if X ∌ u and for every finite subset F of X there is a U ε u such that F ⊆ U. Let fi be the collection of ω-covers of X. We show that the partition relation Ω → [Ω]2/2 holds if, and only if, the partition relation Ω → [Ω]2/3 holds.