Hans-Dietrich O. F. Gronau

Two conjectures of Demetrovics, Furedi, and Katona, concerning partitions

Abstract A concept of descriptive sufficiency is introduced to characterize the subcollections of flats from which the key properties of a matroid can be determined by certain convenient conditions. The concept of descriptive sufficiency is related to the essential flats, and an algorithm is proposed which constructs the erections of an arbitrary matroid in terms of a particular descriptively sufficient subcollection of flats.

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 .

On posets of m -ary words

Abstract The posetBm, n, which consists of the naturally ordered subwords of the cyclic word on length n on an alphabet of m letters, where subwords are obtained by deleting letters, is introduced and studied. This poset is of special interest since it is strongly related to several different structured posets, like Boolean lattices, chains, Higman orders and Kruskal-Katona posets.

Some Ramsey Schur Numbers

The Ramsey Schur number

Ramsey numbers for sets of small graphs

RS(s,t)

An intersection-union theorem for integer sequences

is the smallest

Sperner theory in partially ordered sets

n

Numbers of nonisomorphic drawings for small graphs

such that every 2-colouring of the edges of

The Automorphism Group of the Subsequence Poset Bm, n

K_n

Coverings of the complete (di-)graph with n vertices by complete bipartite (di-)graphs with n vertices I

with vertices