On maximal antichains containing no set and its complement

Abstract Let 1⩽k1⩽k2⩽…⩽kn be integers and let S denote the set of all vectors x = (x1, …, xn with integral coordinates satisfying 0⩽xi⩽ki, i = 1,2, …, n; equivalently, S is the set of all subsets of a multiset consisting of ki elements of type i, i = 1,2, …, n. A subset X of S is an antichain if and only if for any two vectors x and y in X the inequalities xi⩽yi, i = 1,2, …, n, do not all hold. For an arbitrary subset H of S, (i)H denotes the subset of H consisting of vectors with component sum i, i = 0, 1, 2, …, K, where K = k1 + k2 + …kn. |H| denotes the number of vectors in H, and the complement of a vector xϵS is (k1-x1, k2-x2, …, kn -xn). What is the maximal cardinality of an antichain containing no vector and its complement? The answer is obtained as a corollary of the following theorem: if X is an antichain, K is even and |( 1 2 K)X| does not exceed the number of vectors in ( 1 2 K)S with first coordinate different from k1, then ∑ i=0 K i≠ 1 2 K |(i)X| |(i)S| + |( 1 2 K)X| |( 1 2 K-1)S| ⩽1 .

[1]  Daniel J. Kleitman,et al.  Extensions of the Erd¨os-Ko-Rado theorem , 1976 .

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

[3]  D. Lubell.  A Short Proof of Sperner’s Lemma , 1966 .

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

[5]  G. F. Clements.  An Existence Theorem for Antichains , 1977, J. Comb. Theory, Ser. A.

[6]  de Ng Dick Bruijn,et al.  On the set of divisors of a number , 1951 .

[7]  Koichiro Yamamoto.  Logarithmic order of free distributive lattice , 1954 .

[8]  G. F. Clements,et al.  A minimization problem concerning subsets of a finite set , 1973, Discret. Math..

[9]  Gyula O. H. Katona,et al.  Intersection theorems for systems of finite sets , 1964 .

[10]  B. Lindström,et al.  A Generalization of a Combinatorial Theorem of Macaulay , 1969 .

[11]  E. C. Milner.  A Combinatorial Theorem on Systems of Sets , 1968 .

[12]  G. F. Clements.  On existence of distinct representative sets for subsets of a finite set , 1970 .

[13]  D. J. Kleitman.  On an extremal property of antichains in partial orders , 1974 .

[14]  L. D. Meshalkin.  Generalization of Sperner’s Theorem on the Number of Subsets of a Finite Set , 1963 .

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

[16]  L. D. Mesalkin.  A Generalization of Sperner’s Theorem on the Number of Subsets of a Finite Set , 1963 .

[17]  I. Anderson.  On the Divisors of a Number , 1968 .