Partial Steiner triple systems with equal-sized holes
In this elegant paper the authors completely dispose of the existence problem for 3-GDDs of type \(u^ r l^ t\), i.e. of group divisible designs with \(r\) groups of size \(u\) and \(t\) groups of size \(l\), and all blocks of size 3 (here \(u\), \(r\), \(t\) are positive integers). According to their main theorem, such a GDD exists if and only if the following conditions are satisfied: (i) \(u\) is odd, (ii) \(r+ t\) is odd, (iii) if \(r= 1\), \(t\geq u+1\), (iv) if \(r= 2\), then \(t\geq u\), and (v) \(\left(\begin{smallmatrix} t\\ 2\end{smallmatrix}\right)+ rut+ \left(\begin{smallmatrix} r\\ 2\end{smallmatrix}\right) u^ 2\equiv 0\pmod 3\). Their starting point is a theorem in [the first author, \textit{D. G. Hoffman} and \textit{R. Rees}, A new class of group divisible designs with block size three, J. Comb. Theory, Ser. A 59, No. 1, 73-89 (1992; Zbl 0759.05012)] as well as the computational results in [the first author, Small group divisible designs with block size three, J. Comb. Math. Comb. Comput. 14, 153-172 (1993; Zbl 0793.05012)]. Their methods involve the full arsenal of design- theoretical tools, including resolvable GDDs and their variations, transversal designs, and variants of Wilson's fundamental construction.
scientific article; zbMATH DE number 740287