We determine the asymptotics of the largest family of qualitatively 2-independent k-partitions of an n-set, for every k > 2. We generalize a Sperner-type theorem for 2-partite sets of Korner and Simonyi to the k-partite case. Both results have the feature that the corresponding trivial information-theoretic upper bound is tight. The results follow from a more general Sperner capacity theorem for a family of graphs in the sense of our previous work on Sperner theorems on directed graphs.
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