On the asymptotic of lottery numbers

Let $L(n,k,r,p)$ denote the minimum number of $k$-subsets of an $n$-set such that all the $\left(\genfrac{}{}{0px}{}{n}{p}\right)$ $p$-subsets are intersected by one of them in at least $r$ elements. The case $p=r$ corresponds to the covering numbers, while the case $k=r$ corresponds to the Turán numbers. In both cases, there exists a limit of $L(n,k,r,p)\mathrm{/}\left(\genfrac{}{}{0px}{}{n}{r}\right)$ as $n\to \mathrm{\infty }$. We prove the existence of this limit in the general case.