  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
Todor\v{c}evi\'c 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
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 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 V\"a\"an\"anen'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.

Logic;General Topology