Degrees in oriented hypergraphs and sparse Ramsey theory

Let G be an r-uniform hypergraph. When is it possible to orient the edges of G in such a way that every p-set of vertices has some p-degree equal to 0? (The p-degrees generalise for sets of vertices what in-degree and out-degree are for single vertices in directed graphs.) Caro and Hansberg asked if the obvious Hall-type necessary condition is also sufficient.Our main aim is to show that this is true for r large (for given p), but false in general. Our counterexample is based on a new technique in sparse Ramsey theory that may be of independent interest.