We say that a set system ℱ is k-completely hyperseparating if for any vertex v, there are at most k sets in ℱ with intersection {v}. We determine the minimum size of such set systems on an n-element underlying set, generalizing a very recent result for k=2 by Batíková, Kepka, and Nemĕc. We say that ℱ is k-hyperseparating if for any vertex v, there are at most k sets in ℱ such that no other vertex is contained by exactly the same sets out of these k sets. We determine the minimum size of 2-hyperseparating set systems on an n-element underlying set.
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