The topological Baumgartner-Hajnal theorem

Abstract:

Two new topological partition relations are proved. These are \begin{equation*} \omega _{1} \to (top \alpha +1)^{2}_{k}\end{equation*} and \begin{equation*} \mathbb {R} \to (top \alpha +1)^{2}_{k}\end{equation*} for all $\alpha < \omega _{1}$ and all $k< \omega$. Here the prefix “top” means that the homogeneous set $\alpha +1$ is closed in the order topology. In particular, the latter relation says that if the pairs of real numbers are partitioned into a finite number of classes, there is a homogeneous (all pairs in the same class), well-ordered subset of arbitrarily large countable order type which is closed in the usual topology of the reals. These relations confirm conjectures of Richard Laver and William Weiss, respectively. They are a strengthening of the classical Baumgartner-Hajnal theorem.