Transversal designs of block size eight and nine
The author uses a mixture of direct and recursive constructions to almost settle the existence problem for transversal designs with index \(\lambda> 1\) and block sizes \(k= 8\) and \(k= 9\) (with just a few open cases). This is a major result, since it is the first result that goes beyong the work of Hanani who settled the corresponding problem for \(k= 7\), see \textit{H. Hanani} [On transversal designs. Combinatorics, Part 1, Proc. Advanced Study Inst., Breukelen, 42-52 (1974; Zbl 0312.05016)]. The author's proof uses some (generalization of) standard methods, e.g. quasi-difference matrices and \(V(m, t)\) vectors for direct constructions and Wilson-type theorems for recursive constructions, and a new concept (the so-called ``outline transversal design'') which turns out to be a very powerful tool which should have further applications.
scientific article; zbMATH DE number 847626