Another Three Part Sperner Theorem
Jerrold R. Griggs · Studies in Applied Mathematics · 1977
We show that a result of Katona can be made into a three part Sperner theorem which is independent of the best previously known such theorem, in that neither hypothesis implies the other. These three part theorems are stated in terms of a three dimensional rectangular integer lattice L , and give sufficient conditions for F ⊆ L , containing no two points on a line, to be no larger in size than the set of points of middle rank in L . The theorems apply to the more general problem in which L is the product of three symmetric chain orders and F ⊆ L contains no two points equal in two components and ordered in the third.