The number of edge disjoint transitive triples in a tournament

We prove that a tournament with $n$ vertices has more than $0.13n^2(1+o(1))$ edge-disjoint transitive triples. We also prove some results on the existence of large packings of $k$-vertex transitive tournaments in an $n$-vertex tournament. Our proofs combine probabilistic arguments and some powerful packing results due to Wilson and to Frankl and R"odl.