Leírás
The open graph dichotomy for a subset $X$ of the Baire space $\omega^\omega$ states that any open graph on $X$ either contains a large complete subgraph or admits a countable coloring. It is a definable version of the open coloring axiom for $X$ and it generalizes the perfect set property. The focus of this talk is a recent generalization to infinite dimensional directed hypergraphs by Carroy, Miller and Soukup. It is motivated by applications to definable sets of reals, in particular to the second level of the Borel hierarchy.
We show that this infinite dimensional dichotomy holds for all subsets of the Baire space in Solovay's model. Our main results are versions of this theorem for generalized Baire spaces $\kappa^\kappa$ for uncountable regular cardinals $\kappa$. If time permits, we will also look at conditions under which this dichotomy can be strengthened and mention several applications in the setting of generalized Baire spaces.
This is joint work with Philipp Schlicht.
Zoom:
https://us06web.zoom.us/j/4012456659