The existence of \(r\)-large sets of Mendelsohn triple systems
An ordered design with block size three is a pair \((V,{\mathcal B})\) where \(V\) is a set of \(v\) elements and \({\mathcal B}\) is a set of \(v(v-1)\) 3-tuples of distinct elements (blocks) so that for each \(0 \leq i < j < 3\), each ordered pair of distinct elements appears in exactly one block in coordinates \((i,j)\). The 1-rotation of block \((a,b,c)\) is \((b,c,a)\), and its 2-rotation is \((c,a,b)\). The 1- or 2-rotation of an ordered design consists of the 1-rotations (2-rotations, respectively) of each block. An ordered design that is equal to its 1-rotation (and hence to its 2-rotation as well), is equivalent to a Mendelsohn triple system. In this paper, the study of large sets of ordered designs and large sets of Mendelsohn triple systems is generalized to examine partitions of all 3-tuples with distinct elements from \(V\) into \(r\) ordered designs that are each equivalent to a Mendelsohn triple system, and \((v-2-r)/3\) sets each containing an ordered design, its 1-rotation, and its 2-rotation. The paper establishes basic necessary conditions for such a partition to exist, and employs powerful direct and recursive constructions to prove sufficiency for certain parameters, including infinite classes.
scientific article; zbMATH DE number 7358441