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  • ADAM R. DAY (a1) and ANDREW S. MARKS (a2)


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.



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[1]Debs, G. and Raymond, J. S., Filter descriptive classes of Borel functions. Fundamenta Mathematicae, vol. 204 (2009), no. 3, pp. 189213.
[2]Friedman, H. and Stanley, L., A Borel reducibility theory for classes of countable structures, this Journal, vol. 54 (1989), no. 3, pp. 894–914.
[3]Hell, P. and Nešetřil, J., Graphs and Homomorphisms, Series in Mathematics and its Applications, vol. 28, Oxford University Press, Oxford, 2004.
[4]Kechris, A. S., Classical Descriptive Set Theory, Graduate Texts in Mathematics, vol. 156, Springer-Verlag, New York, 1995.
[5]Kechris, A. and Marks, A., Descriptive graph combinatorics, 2015. Preprint available at∼marks.
[6]Kechris, A. S., Solecki, S., and Todorcevic, S., Borel chromatic numbers. Advances in Mathematics, vol. 141 (1999), no. 1, pp. 144.
[7]Louveau, A., On the reducibility order between Borel equivalence relations, Logic, Methodology and Philosophy of Science, IX (Uppsala, 1991) (Prawtiz, D., Skyrms, B., and Westerståhl, D., editors), Studies in Logic and the Foundations of Mathematics, vol. 134, North-Holland, Amsterdam, 1994, pp. 151155.
[8]Marks, A., Slaman, T. A., and Steel, J. R., Martin’s conjecture, arithmetic equivalence, and countable borel equivalence relations, Ordinal Definability and Recursion Theory: The Cabal Seminar, Vol. III (Kechris, A. S., Löwe, B., and Steel, J. R., editors), Cambridge University Press, Cambridge, 2011, pp. 493520.



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