Dorottya Sziraki

Open graphs and hypergraphs on definable subsets of generalized Baire spaces

The open graph dichotomy for a given subset X of the Baire space omega^omega is a generalization of the perfect set property for X which can be viewed as the perfect set version of the Open Coloring Axiom restricted to X. In joint work with P. Schlicht, we extend a theorem of Q. Feng's about the open graph dichotomy for definable subsets of the Baire space to the generalized Baire space kappa^kappa, where kappa is any uncountable cardinal with kappa^(<\kappa)=kappa. More concretely, we show that the kappa-analogue of the open graph dichotomy for all subsets of kappa^kappa which are definable from a kappa-sequence of ordinals is consistent relative to the existence of an inaccessible cardinal above kappa. In the talk, I will sketch a proof of this result. If time allows, I will also report on the progress of possible generalizations of Q. Feng's and our above mentioned theorems for certain definable infinite dimensional hypergraphs. These concern (special cases of) an infinite dimensional version of the open graph dichotomy which was recently introduced by R. Carroy, B.D. Miller and D.T. Soukup.