Counting antichains in finite partially ordered sets

AbstractWe prove that for each integer l > 1 there exists a number r = r(l) > 1 such that every finite poset P of length l − 1 contains an element a satisfying no. of antichains of P containing atotal no. of antichains of P⩾1r For the case l = 2, r = 8.807 will do. A consequence is that every finite distributive lattice L whose poset of join-irreducibles has length one contains a prime ideal I satisfying 19 < |I|/|L| < 89. In the other direction, we show that r(2) cannot be chosen less than 4.3865297