This is Turán's problem. I quote the opening paragraph of A. E. Brouwer and M. Voorhoeve, "Turán theory and the lotto problem", Mathematical Center Tracts 106 (1979), 99-105 = Chapter 7 of A. Schrijver, ed., *Packing and Covering in Combinatorics*, Mathematisch Centrum, Amsterdam, 1979, ISBN 90-6196-180-7.

Let $k,\ell,n\in\mathbb N$ such that $k\le\ell\le n$. We define the *Turán number* $T(n,k,\ell)$ as the smallest number of $k$-subsets of an $n$-set $X$ such that any $\ell$-subset of $X$ contains at least one of these $k$-subsets. For example: $T(7,4,5)=7$. (Take $X=\{0,1,\dots,6\}$; the $4$-subsets are all translates $\pmod7$ of $\{1,2,3,5\}$; this is easily seen to be optimal.) The relation between Turán numbers and covering numbers is discussed in Chapters 4 and 5. The above definition can be formulated in the language of hypergraphs (see Chapter 1) as follows: for a hypergraph $H=(X,\mathcal E)$, let its *stability* number $\beta(H)$ be the maximal cardinality of a stable subset of $H$ (i.e. a set containing no edge). Then $T(n,k,\ell)$ is the minimal number of edges of a $k$-uniform hypergraph $H$ with $n$ vertices such that $\beta(H)\lt\ell$.

P. TURÁN [10] posed the problem of determining $T(n,k,\ell)$. In this section we give some estimates for this number. Notice that $T(n,k,\ell)$ is increasing in $n$ and $k$ and decreasing in $\ell$. Trivially, $T(n,1,\ell)=n-\ell+1$. The numbers $T(n,2,\ell)$ and the corresponding graphs are determined by the following theorem of TURÁN [9].

Combinatorics of Finite Sets, ISBN 0-486-42257-7. $\endgroup$ – Włodzimierz Holsztyński Oct 12 '14 at 2:06