TITLE: Minimal number of n/2-subsets of [n] that contains every d-subset
QUESTION [8 upvotes]: 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!

REPLY [4 votes]: These are called covering designs. See https://www.ccrwest.org/cover.html for references and tables of known values and bounds.