OPEN
Let $f(n)$ be minimal such that there is a tournament (a complete directed graph) on $f(n)$ vertices such that every set of $n$ vertices is dominated by at least one other vertex. Estimate $f(n)$.
Schütte asked Erdős this in the early 1960s.
It is easy to check that $f(1)=3$ and $f(2)=7$. Erdős [Er63c] proved
\[2^{n+1}-1 \leq f(n) \ll n^22^n.\]
Szekeres and Szekeres [SzSz65] proved that $f(3)=19$ and
\[n2^n \ll f(n).\]
