Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2008 / B / 5 / b / ii

Orient each edge independently and uniformly. Fix a $k$-set $A$. An outside vertex beats every member of $A$ with probability $2^{-k}$. For different outside vertices these events depend on disjoint sets of edges, so they are independent. ThusA union bound over all $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ possible sets givesThe complementary event has positive probability, proving a tournament with the required domination property exists. This applies in the intended nonvacuous range $n\ge k$; the asymptotic construction below has $n>k$.