An r-cover-free family is a set system no member is contained in the union of r other members. With r=1, this just means that no set in the system is a subset of another, and they are called Sperner systems.
As for Sperner systems, one asks for the largest cardinality of an r-cover-free family of subsets of an n-element set. Mozhgan presented bounds & proofs from a paper of Erdos, Frankl, and Furedi, and a follow-up paper by Furedi.
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