Several families with incomparability and complementarity conditions

Abstract 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.

[1]  Anthony J. W. Hilton.  On ordered set systems and some conjectures related to the erdös–ko–rado theorem and turán's theorem , 1981 .

[2]  Hans-Dietrich O. F. Gronau,et al.  On maximal antichains containing no set and its complement , 1981, Discret. Math..

[3]  A. J. W Hilton.  INTERSECTING AND NON-UNION ANTICHAINS , 1994 .

[4]  Béla Bollobás,et al.  Multicoloured extremal problems , 2004, J. Comb. Theory, Ser. A.

[5]  E. Sperner.  Ein Satz über Untermengen einer endlichen Menge , 1928 .

[6]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[7]  Jiping Liu,et al.  On a conjecture of Hilton , 2001, Australas. J Comb..

[8]  D. Kleitman,et al.  Proof techniques in the theory of finite sets , 1978 .

[9]  Anthony J. W. Hilton.  An Intersection Theorem for a Collection of Families of Subsets of a Finite Set , 1977 .

[10]  Béla Bollobás,et al.  Sperner Systems Consisting of Pairs of Complementary Subsets , 1973, J. Comb. Theory, Ser. A.

[11]  A. Hilton.  A THEOREM ON FINITE SETS , 1976 .

[12]  P. Seymour.  On incomparable collections of sets , 1973 .

[13]  Peter Frankl,et al.  A generalization of Sperner's theorem , 1981 .

[14]  Benny Sudakov,et al.  Set Systems with Restricted Cross-Intersections and the Minimum Rank of Inclusion Matrices , 2005, SIAM J. Discret. Math..

[15]  I. Anderson.  Combinatorics of Finite Sets , 1987 .

[16]  Yuejian Peng,et al.  On incomparable and uncomplemented families of sets , 2005, Australas. J Comb..