The (n-1)-dimensional Hausdorff measure of any k-antichain in [0,1]^n is at most kn, and for n=2 there exist k-antichains attaining exactly 2k.
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On $k$-antichains in the unit $n$-cube
The (n-1)-dimensional Hausdorff measure of any k-antichain in [0,1]^n is at most kn, and for n=2 there exist k-antichains attaining exactly 2k.