Tur\'{an proved} that, when $r=2$, this limit is\[\frac{1}{2}\left(1-\frac{1}{k-1}\right).\]Erd\H{o}s \cite{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 $k=4$.


References


[Er81] Erd\H{o}s, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.