Stronger Form of an M-Part Sperner Theorem

Katona [7] and Kleitman [11] independently discovered a sharpening of this theorem: Take a partition xl u x2 of X and suppose that there is no pair Ft. Fz Effi, Fl c F2 such that F 2 F 1 cXi for some i = 1, 2. Under this weaker condition the same inequality (1) can be proved. This statement is called the two-part Sperner theorem. Analogously, if the partition X 1 U · · · U XM =X is considered one may exclude the pairs Ft. F 2 E fJi, F 1 c F2 such that F2 F1 c Xi for some i (1 o;; i o;; M). Easy counterexamples show that this condition does not imply (1) even in the case M = 3. [10] and [6] give some additional conditions (for M = 3) ensuring this implication. The exact maximum of lffil under this general condition is unknown. On the other hand Erdos [3] proved that if ffi does not contain I+ 1 different members satisfying F1 c F2 c · · · c Ft+l then lffil does not exceed the sum of the /largest binomial coefficients of order n. A natural combination of the above two conditions is the following one: