Ian T. Roberts

Completely separating systems of k -sets

Dickson (1969) introduced the notion of a completely separating set system. We study such systems with the additional constraint that each set in the system has the same size. Let T denote an n-set. We say that a subset S of T separates i from j if i ∈ S and j ∉ S. A collection of k-sets script c sign is called a (n, k)-separator if, for each ordered pair (i,j) ∈ T × T with i ≠ j, there is a set S ∈ script c sign which separates i from j. Let R(n, k) denote the size of a smallest (n, k)-separator. For n ≥ k(k - 1) we show that R(n, k) = [2n/k]. We also show that R(2,2)≤2m and demonstrate various recursive relationships that are used to determine the exact values of R(n, k) for k ≤ 5.

Antichains and completely separating systems-A catalogue and applications

This paper extends known results on the existence, number and structure of antichains and completely separating systems. Both these structures are classified in several ways, and both an enumeration and listing of each type of object are given in a catalogue, which is described in detail in this paper. The antichain catalogue provides a complete listing of all non-isomorphic antichains on m points for m@?7.

On the existence of regular antichains

A k-regular antichain in the Boolean lattice of subsets is one in which each point occurs in exactly k sets. The existence and construction of k-regular antichains on m points for each positive integer pair (k,m) is determined for all m and most k.

Uniform perfect systems of sets of iterated differences of size 4

A general problem raised by the work of Kreweras and Loeb concerns the existence of partitions of runs of consecutive integers into sets, known as sets of iterated differences, the s(s + 1)/2 elements of which can be expressed as certain linear combinations of s integer valued parameters. In this paper we study the case of four parameters. By examining properties already present in examples with only a single set, we build up the rudiments of an arithmetic of these partitions comparable to that for perfect systems of difference sets and complete permutations.

Regular Perfect Systems of Sets of Iterated Differences

Fors?2, a set {a(i,j):1 ?j?s+1 ?i?s} wherea(1,j), 1? j?s, are some prescribed integers anda(i+1,j) =|a(i,j) ?a(i,j+1)|, for 1?i

Minimizing the weight of the union-closure of uniform families of sets

Abstract The problem of minimising the size, or more generally the weight for certain weight functions, of the union-closure of collections of m distinct i-sets is considered, where m and i are given. Some recent progress toward solving this problem is recorded.