Finding and counting small tournaments in large tournaments

We present new algorithms for counting and detecting small tournaments in a given tournament. In particular, we prove that every tournament on four vertices (there are four) can be detected in $O\left(n^2\right)$ time and counted in $O\left(n^{\omega }\right)$ time where $\omega <2.372$ is the matrix multiplication exponent. We further prove that any tournament on five vertices (there are 12) can be counted in $O\left(n^{\omega +1}\right)$ 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 undirected cliques of order $k-O(\log k)$.