It was proven by Erdos that for any $k$ there exists a $k$-paradoxical tournament, as described here: http://en.wikipedia.org/wiki/Tournament_(graph_theory)#Paradoxical_tournaments.
I am able to show that a $k$-paradoxical tournament exists whenever ${n \choose k}(1-2^{-k})^{n-k} < 1$, but I am unable to show that $n > 2^k k^2 (\ln(2) + \text{o}(1))$ implies this, as the wikipedia article seems to say. I have tried using ${n \choose k} \leq n^k/k!$ and $1-x \leq e^{-x}$ and seem to be getting close, but cannot quite get there.
Any thoughts?