# Topological Ramsey theory of countable ordinals

Mathematical, Foundational and Computational Aspects of the Higher Infinite

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.

This talk is part of the Isaac Newton Institute Seminar Series series.