Show that there is a finite $n_0$ such that any directed graph on $n>n_0$ vertices in which each outdegree is at least $\log_2(n)-\frac{1}{10}\log_2\log_2n$ contains an even simple directed cycle.
I'm looking for hints on the above exercise - I have no ideas so far. I'm not sure what kind of trick I could use to force cycles to be even.
Edit: My thoughts on some of the comments.
Put each vertex randomly into one of two sets - A and B, and let S be the bipartite subgraph containing precisely the edges between A and B. We find that the expected number of vertices in S with outdegree 0 is at most $n{\frac{1}{2}}^{\log_2n-\log_2\log_2n} = \log_2n$. This doesn't suffice. I've thought about deleting vertices of outdegree 0, but I haven't been able to make that work.