A family $\cal F$ of subsets of a finite set is $r$-cover-free if no member of $\cal F$ is contained in the union of $r$ other members of $\cal F$. Let $T(n,r)$ denote the maximum cardinality of an $r$-cover-free family of subsets of an $n$-element set. This concept has arisen independently in several different contexts—information theory, combinatorics, and group testing—under various names (superimposed codes, $ZFD_r$ codes), and bounds on $T(n,r)$ have been rederived several different times.

I almost added to the confusion myself because I rediscovered these objects and was calling them $k$-Sperner sets. Fortunately, before my paper was published, I discovered that my results were already known. See the paper by Miklós Ruszinkó, "On the upper bound of the size of the $r$-cover-free families," J. Combin. Theory Ser. A 66 (1994), 302–310, for a list of the disparate previous papers on the subject, and a proof of the result that for sufficiently large $n$, $\log_2 T(n,r) \le 8n (\log_2 r)/r^2$.