On colouring oriented graphs of large girth

We prove that for every oriented graph D and every choice of positive integers k and $, there exists an oriented graph D* along with a surjective homomorphism 0: V (D*) -> V (D) such that: (i) girth(D*) >= l; (ii) for every oriented graph C with at most k vertices, there exists a homomorphism from D* to C if and only if there exists a homomorphism from D to C; and (iii) for every D-pointed oriented graph C with at most k vertices and for every homomorphism (p : V (D*) V (C) there exists a unique homomorphism f : V (D) V (C) such that (p = f o psi. Determining the oriented chromatic number of an oriented graph D is equivalent to finding the smallest integer k such that D admits a homomorphism to an order -k tournament, so our main theorem yields results on the girth and oriented chromatic number of oriented graphs. While our main proof is probabilistic (hence nonconstructive), for any given l >= 3 and k >= 5, we include a construction of an oriented graph with girth $ and oriented chromatic number k.