Finding and counting small tournaments in large tournaments

We present new algorithms for counting and detecting small tournaments in a given tournament. In particular, it is proved that every tournament on four vertices (there are four) can be detected in $O(n^2)$ time and counted in $O(n^{\omega })$ time where $\omega <2.373$ is the matrix multiplication exponent. It is also proved that any tournament on five vertices (there are $12$) can be counted in $O(n^{\omega +1})$ time. As for lower-bounds, we prove that for almost all $k$-vertex tournaments, the complexity of the detection problem is not easier than the complexity of the corresponding well-studied counting problem for {\em undirected cliques} of order $k-O(\log k)$.