Abstract
I will sketch a game-theoretical proof of the open graph
dichotomy for box-open hypergraphs, and explain how it can be used to
obtain various descriptive-set-theoretical dichotomies at the second
level of the Borel hierarchy. This shows how to generalise these
dichotomies from analytic metric spaces to separable metric spaces by
working under the axiom of determinacy.
If time allows it, I will also discuss some connections between
cardinal invariants and the chromatic number of the graphs at stake.
This is joint work with Benjamin D. Miller and Daniel T. Soukup.