The morphology of partially ordered sets

Abstract

DEFINITION. XC gn is called a set of incomparable elements if A, B E X and A C B imply A = B. Then Sperner’s Theorem -maxx / X 1 = (,J,,), where X ranges over all sets of incomparable elements in .?3,, . The standard proof of Sperner’s theorem (see Harper-Rota [lo]) is based upon two properties of an , unimodality and matching. By unimodality we mean that (3, the number of k-subsets of an n-set, increases to a maximum and then decreases. By matching we mean that the bipartite graph whose vertices are the k-subsets and (k + I)-subsets respectively have a matching in the sense of P. Hall. Tbe conjunction of matching and unimodality imply that an may be partitioned into chains, each chain containing one [n/2]-subset. Any set X of incomparable elements can intersect each chain in at most one element. The injection of X into the [n/2]-subsets thus defined proves the theorem. In 1945 Erdiis improved Sperner’s theorem to give the maximal cardinality of any set XC .G@,, which has no more than k members lying on any chain as C’1’ ( 1 0 [cn+nr,,z,), the sum of the k largest binomial coefficients. In 1951 de Bruijn et al. developed a variant of Sperner’s theorem for the lattice of divisors of n. In the meantime Sperner’s theorem has been reproved in a number of ways, but by far the most elegant proof is due to Lubell. In 1967 Rota pointed out the possibility of extending Sperner’s theorem to the lattice of partitions of an n-set. The author’s plan in attacking Rota’s conjecture has been to build up an arsenal of theoretical weapons sufficient