Closed ordinal Ramsey numbers below $\omega^\omega$

Since the 1950s, many versions of the partition calculus and arrow notation, introduced by Erdős and Rado, were studied. One such variant, introduced by Baumgartner and recently studied by Caicedo and Hilton, is the closed ordinal Ramsey number. For this variant, we require our homogeneous subset to be both order-isomorphic and homeomorphic to a given ordinal.

In the talk we present an approach with which to tackle this flavour of partition calculus, and if time permits prove some results. The talk is elementary and self-contained.