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 divide  ℓ , any large enough oriented graph  G with δ + ( G ) , δ − ( G ) ≥ ⌊ | V ( G ) | / k ⌋ + 1 contains a directed cycle of length  ℓ . We prove this conjecture asymptotically for the case when  ℓ is large enough compared to k and k ≥ 7 . The case when  k ≤ 6 was already settled asymptotically by Kelly, Kühn and Osthus.