Let $d , n$ be positive integers such that $d < n/2$. Consider collections $\mathcal{F}$ consisting of subsets of $[n] = \{1,2,\ldots, n\}$ of size $n/2$. Question: what is the minimal size of a collection $\mathcal{F}$, such that every size-$d$ subset of $[n]$ is contained in \textit{at least} one set in $\mathcal{F}$ ?

I did searches in literature and if the "at least" above is changed to "exactly one" then it is called Steiner designs and is considered a hard problem. Just wondering if the above version is easier...

I had some initial idea but got stuck on how to make it work: say $n = 2^k$, then we can regard $[n]$ as $\mathbb{F}_2^k$, and select all hyperplanes. But I don't know if such collection satisfies the condition ... any ideas are appreciated!