The open dihypergraph dichotomy and the second level of the Borel hierarchy

TL;DR

This paper generalizes classical dichotomy theorems related to the second level of the Borel hierarchy using dihypergraph concepts, linking set-theoretic and combinatorial properties under determinacy axioms.

Contribution

It introduces an $eth_0$-dimensional open dihypergraph dichotomy framework that unifies several known results and extends them to broader classes of metric spaces.

Findings

Several dichotomy theorems are special cases of the dihypergraph dichotomy.

Under determinacy, results extend from analytic to separable metric spaces.

Connections between cardinal invariants and dihypergraph chromatic numbers are established.

Abstract

We show that several dichotomy theorems concerning the second level of the Borel hierarchy are special cases of the $\aleph_0$-dimensional generalization of the open graph dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under the axiom of determinacy, we obtain the generalizations of these results from analytic metric spaces to separable metric spaces. We also consider connections between cardinal invariants and the chromatic numbers of the corresponding dihypergraphs.