Minimal witnesses to k unlabelled posets of size n have size at most nk-n (with a 3/2(n+1) example for k=3), and no linear bound n+k+c exists.
The mathematics of causal sets , pages 339–365
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Sizes of witnesses in Covtree
Minimal witnesses to k unlabelled posets of size n have size at most nk-n (with a 3/2(n+1) example for k=3), and no linear bound n+k+c exists.