Minimal coverings and maximal packings of $(k-1)$-subsets by $k$-subsets

N. N. Kuzyurin

Matematičeskie zametki, Tome 21 (1977) no. 4, pp. 565-571

Résumé

This paper studies the asymptotic behavior of functions $M(n,k,k-1,\lambda)$ and $m(n,k,k-1,\lambda)$, equal to the respective cardinalities of the minimal $\lambda$-covering and maximal $\lambda$-packing of all $(k-1)$-subsets of the $n$-element set of its $k$-subsets. It is shown that, if sequence $k=k(n)$ is such that $k(n)/n\to0$ as $n\to\infty$ then $m(n,k,k-1,\lambda)\sim\lambda\cdot\bigl({n\atop k-1}\bigr)\cdot k^{-1}$, and $k(n)/\sqrt n\to0$ as $n\to\infty$, then $M(n,k,k-1,1)\sim\lambda\cdot\bigl({n\atop k-1}\bigr)\cdot k^{-1}$. A consequence of these results is the validity of the Erdös–Hanani conjecture concerning the asymptotic behavior of functions $M(n,k,k-1,1)$ and $m(n,k,k-1,1)$.