Orienting Borel Graphs

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 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 $\mathbf{\Sigma}^1_2$ in the codes, in stark contrast to the case of smooth relations.