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 Todor\v{c}evi\'c to investigate 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. 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 functions which are $\sigma$-continuous with closed pieces.Comment: 115 pages, 11 figures. Added new results in Section 6.2.2 which strengthen and replace the results in Section 6.3 of the previous version. Improved results in Section 5.3. Various other minor corrections. Comments are welcom

Set Theory

This stimulating workshop exposed some of the most exciting recent develops in set theory, including major new results about the proper forcing axiom, stationary reflection, gaps in P(ω)/Fin, iterated forcing, the tree property, ideals and colouring numbers, as well as important new applications of set theory to C*-algebras, Ramsey theory, measure theory, representation theory, group theory and Banach spaces