Abstract

Recall that the Ramsey number R(n, m) is the least k such that, whenever the edges of the complete graph on k vertices are coloured red and blue, then there is either a complete red subgraph on n vertices or a complete blue subgraph on m vertices – for example, R(4, 3) = 9. This generalises to ordinals: given ordinals $lpha$ and $eta$, let $R(lpha, eta)$ be the least ordinal $gamma$ such that, whenever the edges of the complete graph with vertex set $gamma$ are coloured red and blue, then there is either a complete red subgraph with vertex set of order type $lpha$ or a complete blue subgraph with vertex set of order type $eta$ —- for example, $R(omega 1, 3) = omega 1$. We will prove the result of Erdos and Milner that $R(lpha, k)$ is countable whenever $lpha$ is countable and k is finite, and look at a topological version of this result. This is joint work with Andres Caicedo.