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We study maximal families A of subsets of [n] = {1, 2,. .. , n} such that A contains only pairs and triples and A ⊆ B for all {A, B} ⊆ A, i.e. A is an antichain. For any n, all such families A of minimum size are determined. This is equivalent to finding all graphs G = (V, E) with |V | = n and with the property that every edge is contained in some triangle… (More)

This paper deals with completion of partial latin squares L = (l ij) of order n with k cyclically generated diagonals (l i+t,j+t = l ij + t if l ij is not empty; with calculations modulo n). There is special emphasis on cyclic completion. Here, we present results for k = 2,. .. , 7 and odd n ≤ 21, and we describe the computational method used… (More)

In the present paper we will prove that every partial latin square L = (l ij) of odd order n with 2 cyclically generated diagonals (l i+t,j+t = l ij +t if l ij is not empty; with calculations modulo n) can be cyclically completed.

- Ian T. Roberts, Sue D’Arcy, Judith Egan, Martin Grüttmüller, M. Grüttmüller
- 2005

The cardinality of the minimal pairwise balanced designs on v elements with largest block size k is denoted by g (k) (v). It is known that 31 ≤ g (4) (18) ≤ 33. In this paper we show that g (4) (18) = 31.

In this talk, we construct, by using Skolem-type and Rosa-type sequences, cyclically indecomposable twofold triple systems T S2(v) for all admissible orders. We also investigate exhaustively the cyclically indecomposable triple systems T S λ (v) for λ = 2, v ≤ 33 and λ = 3, v ≤ 21 and we identify the decomposable ones. Further, we investigate exhaustively… (More)

- Martin Grüttmüller, Ian T. Roberts, Sue D’Arcy, Judith Egan
- 2005

The minimum number of blocks having maximum size precisely four that are required to cover, exactly λ times, all pairs of elements from a set of cardinality v is denoted by g