A CONTINUOUS ANALOG OF SPERNERS THEOREM

One of the best-known results of extremal combinatorics is Sperner's t heorem, which asserts that the maximum size of an antichain of subsets of an n-element set equals the binomial coefficient [GRAPHICS] that i s, the maximum of the binomial coefficients. In the last twenty years, Sperner's theorem has been generalized to wide classes of partially o rdered sets. It is the purpose of the present paper to propose yet ano ther generalization that strikes in a different direction. We consider the lattice Mod(n) of linear subspaces (through the origin) of the ve ctor space R(n). Because this lattice is infinite, the usual methods o f extremal set theory do not apply to it. It turns out, however, that the set of elements of rank of the lattice Mod(n), that is, the set of all subspaces of dimension k of R(n), or Grassmannian, possesses an i nvariant measure that is unique up to a multiplicative constant. Can t his multiplicative constant be chosen in such a way that an analogue o f Sperner's theorem holds for Mod(n), with measures on Grassmannians r eplacing binomial coefficients? We show that there is a way of choosin g 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 Spe rner's theorem can be proven. The methods of the present note indicate that other results of extremal set theory may be generalized to the l attice Mod(n) by similar reasoning. (C) 1997 John Wiley & Sons, Inc.