Embedding cycles of given length in oriented graphs

Kelly, Kühn and Osthus conjectured that for any ℓ ≥ 4 and the smallest number k ≥ 3 that does not divideℓ, any large enough oriented graphG with δ (G), δ (G) ≥ {pipe} V (G) {pipe} / k + 1 contains a directed cycle of lengthℓ. We prove this conjecture asymptotically for the case whenℓ is large enough compared to k and k ≥ 7. The case whenk ≤ 6 was already settled asymptotically by Kelly, Kühn and Osthus.