On the f-vectors of Cutsets in the Boolean Lattice

A cutset in the poset 2[n], of subsets of {1, …, n} ordered by inclusion, is a subset of 2[n] that intersects every maximal chain. Let 0⩽α⩽1 be a real number. Is it possible to find a cutset in 2[n] that, for each 0⩽i⩽n, contains at most α (ni) subsets of size i? Let α(n) be the greatest lower bound of all real numbers for which the answer is positive. In this note we prove the rather surprising fact that limn→∞ α(n)=0.