Finite cutsets and finite antichains

Abstract

An ordered set (P,≤) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x} ∪ Fx meets every maximal chain of (P,≤). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardinality less than n. We specify n for m < w. Indeed, n-1=m= width P for m=1,2,n=5 if m=3 and n⩾ℵ₁ if m ≥4. With the added hypothesis that every bounded chain has a supremum and infimum in P, it is shown that for 4⩽m⩽ℵ₀, n=ℵ₀. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain.