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. It is a strong version of the open coloring axiom for $X$ that was introduced by Todor\v{c}evi\'c and Feng to study definable sets of reals. We first show that its recent infinite dimensional generalization by Carroy, Miller and Soukup holds for all subsets of the Baire space in Solovay's model, extending a theorem of Feng from dimension $2$. Our 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 L\'evy 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 extends to all relatively box-open directed hypergraphs on any subset of ${}^\kappa\kappa$ that is definable from a $\kappa$-sequence of ordinals. We derive several applications to definable subsets of generalized Baire spaces, among them variants of the Hurewicz dichotomy that characterizes subsets of $K_\sigma$ sets, an asymmetric version of the Baire property, an analogue of the Kechris-Louveau-Woodin dichotomy that characterizes when two disjoint sets can be separated by an $F_\sigma$ set, the determinacy of V\"a\"an\"anen's perfect set game for all subsets of ${}^\kappa\kappa$, and an analogue of the Jayne-Rogers theorem that characterizes the functions which are $\sigma$-continuous with closed pieces.