Games of Chains and Cutsets in the Boolean Lattice II

B. Bájnok and S. Shahriari proved that in 2, the Boolean lattice of order n, the width (the maximum size of an antichain) of a non-trivial cutset (a collection of subsets that meets every maximal chain and does not contain ∅ or [n]) is at least n− 1. We prove that, for n ≥ 5, in the Boolean lattice of order n, given ⌈n2 ⌉− 1 disjoint long chains, we can enlarge the collection to a cutset of width n − 1. However, there exists a collection of ⌈n2 ⌉ long chains that cannot be so extended. Introduction and the result Let [n] = {1, 2, . . . , n} and 2[n] be the partially ordered set of all the subsets of [n] ordered by inclusion. In other words, 2[n] is the Boolean lattice of order n. A long chain in 2[n] is a collection of n − 1 subsets A1 ⊂ A2 ⊂ · · · ⊂ An−1 such that |Ai| = i. A long chain in 2[n] is just a maximal chain minus the empty set and the full set. Let C ⊆ 2[n]. C is a cutset in 2[n] if it intersects every long chain non-trivially. In other words, if we consider the Hasse diagram of 2[n] as a graph, then a cutset is a collection of subsets whose removal would place the empty set and the full set in two different connected components. (Note that usually a cutset is defined to be a collection of subsets that meets every maximal (as opposed to long) chain [3, 4]. The disadvantage of the usual definition is that any collection of subsets that includes the empty set or the full set will (trivially) be a cutset.) Any collection of pairwise incomparable elements in a poset is called an anti-chain, and the size of the largest anti-chain in a poset is the width of the poset. Theorem 1 of [2] together with Dilworth’s theorem (Theorem 3.2.1 of [1]) gives the following: Proposition 1 ([2]) Let C be a cutset in the Boolean lattice of order n, with n ≥ 2. Then the width of C is greater or equal to n − 1. Furthermore, for n ≥ 3 there does exists cutsets of width n − 1 in the Boolean lattice of order n. Partially supported by an undergraduate summer research grant from the Southern California Edison Company. Current Address: Dept. of Math., U. of Washington, Seattle WA 98195. Partially supported by a grant from the Edison Company.