Let us consider a set $A$ containing $n$ elements. What is the minimum number of subsets of $A$ of order $k\geq 2$ such that, for every $(x, y)\in A^2$, at least one those subsets contains both $x$ and $y$ as its elements?

We know that the number of unordered pairs of elements of $A$ is
$$\binom{n}{2},$$
but the *minimum* number of such subsets is definitely smaller. For $k = 5$, the subset $\{a,b,c,d,e\}$ already yields $10$ of these pairs. And if $n = k$, then the answer is trivially $1$.

Is there a way to compute the minimum number of subsets that satisfy this condition, given $n$ and $k$?