We investigate when a Borel graph admits a (Borel or measurable) orientation with outdegree bounded by k for various cardinals k. We show that for a probability measure preserving (p.m.p) graph G, a measurable orientation can be found when k is larger than the normalized cost of the restriction of G to any positive measure subset. Using an idea of Conley and Tamuz, we can also find Borel orientations of graphs with subexponential growth; however, for every k we also find graphs which admit measurable orientations with outdegree bounded by k but no such Borel orientations. Finally, for special values of k we bound the projective complexity of Borel k-orientability for graphs and graphings of equivalence relations. It follows from these bounds that the set of equivalence relations admitting a Borel selector is Sigma(1)(2) in the codes, in stark contrast to the case of smooth relations.
展开▼
