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arXiv · 1604.02228

Jump operations for Borel graphs

Adam R. Day·Andrew S. Marks
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Abstract

We investigate the class of bipartite Borel graphs organized by the order of Borel homomorphism. We show that this class is unbounded by finding a jump operator for Borel graphs analogous to a jump operator of Louveau for Borel equivalence relations. The proof relies on a non-separation result for iterated Frechet ideals and filters due to Debs and Saint Raymond. We give a new proof of this fact using effective descriptive set theory. We also investigate an analogue of the Friedman-Stanley jump for Borel graphs. This analogue does not yield a jump operator for bipartite Borel graphs. However, we use it to answer a question of Kechris and Marks by showing that there is a Borel graph with no Borel homomorphism to a locally countable Borel graph, but each of whose connected components has a countable Borel coloring.

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BibTeXRIS

Adam R. Day, Andrew S. Marks. 2016-04-08. Jump operations for Borel graphs. https://doi.org/10.1017/jsl.2017.45

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