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Topological Ramsey theory of countable ordinals

Presented by: 
J Hilton University of Leeds
Date: 
Saturday 10th October 2015 - 13:00 to 13:55
Venue: 
INI Seminar Room 1
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 $\alpha$ and $\beta$, let $R(\alpha, \beta)$ 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 $\alpha$ or a complete blue subgraph with vertex set of order type $\beta$ --- for example, $R(\omega + 1, 3) = \omega + 1$. We will prove the result of Erdos and Milner that $R(\alpha, k)$ is countable whenever $\alpha$ is countable and k is finite, and look at a topological version of this result. This is joint work with Andres Caicedo.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons