The Second Sch\"utte Number is Seven

A tournament has property $S_k$ when every set of $k$ vertices is dominated by a common vertex outside it, and $f(k)$ denotes the least order of a tournament with that property. The values $f(1) = 3$ and $f(2) = 7$ are classical; we prove the second here with both bounds machine-checked. The upper bound is realised by the Paley tournament on $Z/7$, in which $i$ beats $j$ when $j - i$ is a non-zero quadratic residue. The lower bound is an exhaustive search over all $2^15$ tournaments on six vertices. What makes such a search exhaustive is an encoding rather than an argument: a tournament is stored as the bits of its upper triangle, so antisymmetry becomes a property of the representation, and the numbers below $2^15$ enumerate the tournaments on six labelled vertices exactly once each.