Topological Ramsey numbers and countable ordinals

We study the topological version of the partition calculus in the setting of countable ordinals. Let α and β be ordinals and let k be a positive integer. We write β →top (α, k)² to mean that, for every red-blue coloring of the collection of 2-sized subsets of β, there is either a red-homogeneous set homeomorphic to α or a blue-homogeneous set of size k. The least such β is the topological Ramsey number Rtop(α, k). We prove a topological version of the Erdős-Milner theorem, namely that Rtop(α, k) is countable whenever α is countable. More precisely, we prove that Rtop(ωωβ, k + 1) ≤ ωωβ·k for all countable ordinals β and finite k. Our proof is modeled on a new easy proof of a weak version of the Erdős-Milner theorem that may be of independent interest. We also provide more careful upper bounds for certain small values of α, proving among other results that Rtop(ω + 1, k + 1) = ωk + 1, Rtop (α, k) < ωω whenever α < ω2, Rtop(ω², k) ≤ ωω and Rtop(ω² + 1, k + 2) ≤ ωω·k + 1 for all finite k.