Every poset has a central element

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.