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We consider subsets, $A$, of the unit $n$-cube $[0,1]^n$ that satisfy \\[ \\text{card}(A \\cap C) \\le k, \\, \\text{ for all chains } \\, C \\subset [0,1]^n \\, , \\] where $k$ is a fixed positive integer. We refer to such a set $A$ as a $k$-antichain. 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