Every digraph on n vertices with minimum out-degree at least n/k contains a directed cycle of length at most k, provided the independence number is at most (k+1)/2.
Proof of the Caccetta-H¨ aggkvist con jecture for oriented graphs with positive minimum out-degree and of independence number two
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Proof of the Caccetta-Haggkvist conjecture for digraphs with small independence number
Every digraph on n vertices with minimum out-degree at least n/k contains a directed cycle of length at most k, provided the independence number is at most (k+1)/2.