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1 results

  • MEASURABLE PERFECT MATCHINGS FOR ACYCLIC LOCALLY COUNTABLE BOREL GRAPHS

  • CLINTON T. CONLEY, BENJAMIN D. MILLER
  • Journal:
    The Journal of Symbolic Logic / Volume 82 / Issue 1 / March 2017
    Published online by Cambridge University Press:
    21 March 2017, pp. 258-271
    Print publication:
    March 2017

  • We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating μ-hyperfinite equivalence relations admit μ-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a counterexample in the locally countable case.