Jump operations for Borel graphs

TL;DR

This paper explores the complexity of bipartite Borel graphs using jump operators and effective descriptive set theory, revealing unboundedness and limitations of certain analogues, and addressing a question about Borel homomorphisms.

Contribution

It introduces a jump operator for Borel graphs, provides a new proof of a non-separation result, and analyzes an analogue of the Friedman-Stanley jump, advancing understanding of Borel graph homomorphisms.

Findings

The class of bipartite Borel graphs is unbounded under Borel homomorphism order.

A new proof of a non-separation result for iterated Frechet ideals and filters is provided.

An analogue of the Friedman-Stanley jump does not serve as a jump operator for bipartite Borel graphs.

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.