It is proved that there exists a constant δ, 1 2 > δ > 0, such that in every finite partially ordered set there is an element such that the fraction of order ideals containing that element is between δ and 1-δ. It is shown that δ can be taken to be at least (3-log2 5) 4≊0.17. This settles a question asked independently by Colburn and Rival, and Rosenthal. The result implies that the information-theoretic lower bound for a certain class of search problems on partially ordered sets is tight up to a multiplicative constant.
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in part by NSF under