Andrés Caicedo: Topological partition calculus of countable ordinals

Thursday, April 14, 2016; 4:00-5:30 PM, in East Hall 2866.

This is joint work with Jacob Hilton. We considered the topological version of the partition calculus in the setting of countable ordinals: Given ordinals $\alpha,\beta_0,\beta_1$, we say that $\alpha\to_{top}(\beta_0,\beta_1)^2$ iff for any 2-coloring of the edges of the complete graph on $\alpha$ vertices, for some color $i$, there is a complete monochromatic graph in color $i$ whose set of vertices is homeomorphic to $\beta_i$. If we insist that $\alpha,\beta_0$ are countable and that $\beta_0>\omega$, then $\beta_1$ must be finite (even if we only require order-types rather than homeomorphisms). On the other hand, we have proved that for any countable $\beta_0$ and finite $\beta_1$, we can find a countable $\alpha$ such that $\alpha\to_{top}(\beta_0,\beta_1)^2$. This is a topological version of the Erdös-Milner theorem. Our arguments provide explicit bounds. I will discuss some of these results.

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