Another Note on Sperner's Lemma
David A. Drake · Canadian Mathematical Bulletin · 1971
Let Q be a finite partially ordered (by ≤) set with universal bounds O, I. The height function h of Q is defined by the rule: h(x) is the maximum length of a chain from O to x. Let h(I)=n. Suppose that for each k≥0, there exist positive integers a(k) and b(k) such that all elements of height k (i) are covered by a(k) elements of height k+1; (ii) cover b(k) elements of height k—1. Then we call Q a U-poset. Call a subset S of a partially ordered set an antichain if no two elements of S are comparable.