On the f-vectors of Cutsets in the Boolean Lattice

A cutset in the poset 2, of subsets of {1, . . . , n} ordered by inclusion, is a subset of 2 that intersects every maximal chain. Let 0 ≤ α ≤ 1 be a real number. Is it possible to find a cutset in 2 that, for each 0 ≤ i ≤ n, contains at most α ( n i ) 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. Introduction and the Result Let [n] = {1, 2, . . . , n} be a set with n elements and let 2[n] denote the poset of all subsets of [n] ordered by inclusion. 2[n] is called the Boolean lattice of order n. We will denote the subsets of size i of [n] by ([n] i ) . Let C ⊆ 2[n]. C is called a cutset in 2[n] if C intersects every maximal chain in 2[n]. C is a non-trivial cutset if it is a cutset that does not contain ∅ or [n]. Given C ⊆ 2[n], the f -vector (or profile) of C is an n+ 1-tuple (f0, f1, . . . , fn) where fi = |C ∩ ([n] i ) |. We are interested in non-trivial cutsets which contain a fixed percentage of the subsets of each possible size. For each n, clearly there exists a greatest lower bound for the percentages α such that it is possible to construct a cutset that contains α (n i ) elements of ([n] i ) for each 1 ≤ i ≤ n− 1. In this note we are interested in the asymptotic behavior of this α as n → ∞. Thus we define α(n) = inf{0 < α ≤ 1 | (0, ⌊α (