The subject of topological partition relations provides answers to questions of the following form: Given a topological space X and a subspace Y, is it possible to reduce any given coloring of the pairs of elements of X to a simpler coloring, by passing to a subspace homeomorphic to Y?

I will first present a survey of topological partition relations for countable ordinals with the order topology. In many instances it is useful to represent countable ordinals using families of fi nite sets. I will describe how to obtain such representations; and will present results from a joint project with William Weiss, where we obtain topological partition relations for ordinals below ω2 with the order topology.