| Abstract: | 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. |
