Generalized Steiner triple systems with group size \(g=7,8\). (Q2715983)

Generalized Steiner triple systems GS\((2,3,n,g)\) can be described as group divisible designs GDD\((g^n)\) (\(n\) groups of size \(g\)), in which all blocks have three elements, and any two intersecting blocks intersect at most two common groups of the GDD. Fix \(g \geq 2\), and denote by \(T_g\) the set of \(n\), for which there exists GS\((2,3,n,g)\). Furthermore, denote by \(B_g\) the set of \(n \geq g+2\), where \((n-1)g\) is even, and \(n(n-1)g^2\) is divisible by three, and let \(M_g\) be the subset of \(T_g\) consisting of \(n \leq 9g + 158\). By earlier results, \(T_g \subseteq B_g\), and \(T_g = B_g\) when \(2 \leq g \leq 6\) and \(g=9\), with the exception \((g,n) = (2,6)\). The authors show that \(M_g \subseteq T_g\) implies \(T_g = B_g\), and this result helps them to prove \(B_7 = T_7\) and \(B_8 = T_8\). The proofs require a lot of (product and holey) constructions, and involve disjoint incomplete Latin squares, difference matrices, resolvable GDD and ad hoc arguments aided by computers.