OPEN - $500
Determine, for any $k>r>2$, the value of
\[\frac{\mathrm{ex}_r(n,K_k^r)}{\binom{n}{r}},\]
where $\mathrm{ex}_r(n,K_k^r)$ is the largest number of $r$-edges which can placed on $n$ vertices so that there exists no $K_k^r$, a set of $k$ vertices which is covered by all $\binom{k}{r}$ possible $r$-edges.
Turán proved that, when $r=2$, this limit is
\[\frac{1}{2}\left(1-\frac{1}{k-1}\right).\]
Erdős
[Er81] offered \$500 for the determination of this value for any fixed $k>r>2$, and \$1000 for 'clearing up the whole set of problems'.
See also [500] for the case $r=3$ and $k=4$.
