The open dihypergraph dichotomy for generalized Baire spaces and its applications

The open graph dichotomy for a subset $X$ of the Baire space ${}^{\omega }\omega$ states that any open graph on $X$ either admits a coloring in countably many colors or contains a perfect complete subgraph. This strong version of the open graph axiom for $X$ was introduced by Feng and Todorčević to study definable sets of reals. We first show that its recent generalization to infinite dimensional directed hypergraphs by Carroy, Miller and Soukup holds for all subsets of the Baire space in Solovay’s model, extending a theorem of Feng in dimension $2$. The main theorem lifts this result to generalized Baire spaces ${}^{\kappa }\kappa$ in two ways. (1) For any regular infinite cardinal $\kappa$, the following holds after a Lévy collapse of an inaccessible cardinal $\lambda >\kappa$ to ${\kappa }^{+}$. Suppose that $H$ is a $\kappa$-dimensional box-open directed hypergraph on a subset of ${}^{\kappa }\kappa$ such that $H$ is definable from a $\kappa$-sequence of ordinals. Then either $H$ admits a coloring in $\kappa$ many colors or there exists a continuous homomorphism from a canonical large directed hypergraph to $H$. (2) If $\lambda$ is a Mahlo cardinal, then the previous result extends to all box-open directed hypergraphs on any subset of ${}^{\kappa }\kappa$ that is definable from a $\kappa$-sequence of ordinals. This approach is applied to solve several problems about definable subsets of generalized Baire spaces. For instance, we obtain variants of the Hurewicz dichotomy that characterizes subsets of $K_{\sigma }$ sets, strong variants of the Kechris-Louveau-Woodin dichotomy that characterizes when two disjoint sets can be separated by an $F_{\sigma }$ set, the determinacy of Väänänen’s perfect set game for all subsets of ${}^{\kappa }\kappa$, an asymmetric version of the Baire property and an analogue of the Jayne-Rogers theorem that characterizes $G_{\delta }$-measurable functions.