An incomplete transversal design ITD\((k,v,h)\) is a design with \(kv\) points partitioned into \(k\) disjoint \(v\)-subsets called groups. The blocks of the design contain exactly one element from each group. The word ‘incomplete’ refers to the presence of a hole \(H\), which contains \(h\) elements from each group. Any two points from different groups appear in exactly one block together, unless they are both in \(H\) in which case they never appear together. An ITD\((k,v,h)\) is said to be idempotent if there exists a set of \(v-h\) blocks containing between them every point once except the points in \(H\).
A necessary condition for an ITD\((k,v,h)\) to exist is that \(v\geq(k-1)h\). In this paper this condition is shown to be sufficient when \(k=5\) except when \(h=1\) and \(v=6\) and possibly when \(h=1\) and \(v=10\). The second author has previously shown [Australas. J. Comb. 12, 193-199 (1995; Zbl 0836.05013)] the same result under the extra restriction \(h>53\). The present proof uses direct constructions from quasi-difference matrices as well as some recursive constructions.
Although incomplete mutually orthogonal Latin squares (IMOLS) are mentioned in the title there seems to be no mention of Latin squares of any sort anywhere else in the paper! This is quite incredible as they are presumably a major part of the motivation for this work. An ITD\((k,v,h)\) can be interpreted as a set of \(k-2\) IMOLS of order \(v\) with a hole of order \(h\). If the ITD is idempotent the IMOLS can be written so that they are idempotent (except that the entries in the hole are missing). Hence, although they allude to it only in their title, the authors have settled the existence question for sets of 3 idempotent IMOLS except for the case \(v=10\), \(h=1\). This unsolved case is equivalent to the famous open question of the existence of 3 MOLS of order 10.