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.Comment: 20 pages, 3 figure