Several families with incomparability and complementarity conditions

Two sets are weakly incomparable if neither properly contains the other; they are strongly incomparable if they are unequal and neither contains the other. Two families A and B of sets are weakly (or strongly) incomparable if no set in one of A and B is weakly (or strongly) comparable to a set in the other. A family A of sets is uncomplemented if A contains no subset and its complement. We show that the following two statements are equivalent (as either can be deduced from the other). 1. If A 1 , … , A t are uncomplemented, mutually weakly incomparable families of subsets of an n-set, then | A 1 | + ⋯ + | A t | ⩽ m a x { 2 n − 1 , t n ⌊ n 2 ⌋ + 1 } . 2. If A 1 , … , A t are uncomplemented, mutually strongly incomparable families of subsets of an n-set, then | A 1 | + ⋯ + | A t | ⩽ 2 n − 1 . Both these relate to a conjecture of Hilton made in 1976 reported in a Math. Review article by D. J. Kleitman. We also show that if A 1 , … , A t are mutually weakly incomparable families of subsets of an n-set, and if they are mutually uncomplemented (i.e. if A ∈ A j, then A ̄ ∉ A i if i ≠ j) then | A 1 | + ⋯ + | A t | ⩽ m a x { 2 n , t n ⌊ n 2 ⌋ + 1 } . We also show that if A 1 , … , A t are t mutually uncomplemented Sperner families of subsets of an n-set, then | A 1 | + ⋯ + | A t | ⩽ t n ⌊ n 2 ⌋ , if n is odd. n n 2 + ( t − 1 ) n n 2 + 1 , if n is even.