A continuous analogue of Sperner's theorem

Abstract

One of the best-known results of extremal combinatorics is Sperner's theorem, which asserts that the maximum size of an antichain of subsets of an n-element set equals the binomial coefficient (n/(n/2)), that is, the maximum of the binomial coefficients. In the last twenty years, Sperner's theorem has been generalized to wide classes of partially ordered sets. It is the purpose of the present paper to propose yet another generalization that strikes in a different direction. We consider the lattice Mod(n) of linear subspaces (through the origin) of the vector space Rn. Because this lattice is infinite, the usual methods of extremal set theory do not apply to it. It turns out, however, that the set of elements of rank k of the lattice Mod(n), that is, the set of all subspaces of dimension k of Rn, or Grassmannian, possesses an invariant measure that is unique up to a multiplicative constant. Can this multiplicative constant be chosen in such a way that an analogue of Sperner's theorem holds for Mod(n), with measures on Grassmannians replacing binomial coefficients? We show that there is a way of choosing such constants for each level of the lattice Mod(n) that is natural and unique in the sense defined below and for which an analogue of Sperner's theorem can be proven. The methods of the present note indicate that other results of extremal set theory may be generalized to the lattice Mod(n) by similar reasoning. © 1997 John Wiley & Sons, Inc.