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JUMP OPERATIONS FOR BOREL GRAPHS

Published online by Cambridge University Press:  01 May 2018

ADAM R. DAY  and
ANDREW S. MARKS
ADAM R. DAY
Affiliation:
SCHOOL OF MATHEMATICS, STATISTICS AND OPERATIONS RESEARCH VICTORIA UNIVERSITY OF WELLINGTON WELLINGTON, NEW ZEALANDE-mail:adam.day@vuw.ac.nz
ANDREW S. MARKS
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA, LOS ANGELES LOS ANGELES, CA, USAE-mail:marks@math.ucla.edu
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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 nonseparation result for iterated Fréchet 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.

Keywords

03E1503D60descriptive set theoryeffective descriptive set theoryBorel graph combinatoricsBorel graph coloringFriedman-Stanley jumpgraph homomorphismFrechet ideals
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 1 , March 2018 , pp. 13 - 28
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

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