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

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arxiv 1803.03205 v1 pith:7YK6KBT6 submitted 2018-03-08 math.LO math.CO

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

classification math.LO math.CO
keywords dichotomyborelhierarchylevelmetricopensecondspaces
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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.

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