Some results on transverse Steiner quadruple systems of type \(g^tu^1\). (Q2810175)

One possible analogue of group divisible designs defined originally by Hanani is a transverse Steiner quadruple system of type \(s_1^{u_1} \dots s_m^{u_m}\) (TSQS(\(s_1^{u_1} \dots s_m^{u_m}\))). It is a triple \((X,\mathcal H,\mathcal B)\) where \(X\) is a set of ``points'', \(\mathcal H\) is a partition of \(X\) into \(u_i\) ``groups'' or ``holes'' with \(s_i\) points, \(i=1,2,\dots,m\), and \(\mathcal B\) is a set of \(4\)-subsets of \(X\) called ``blocks'' such that each block intersects a group in at most one point, and any transverse \(3\)-subset of \(X\) is contained in exactly one block.NEWLINENEWLINE After reviewing the known results on TSQS(\(s^u\)) and TSQS(\(s^ut^1\)), the author presents many new constructions for TSQS(\(s^ut^1\)), in particular, when there are several groups of size \(2\) and one group of larger size, and also for TSQSs of type \(4^n2^1\). The existence problem for TSQSs of latter type, as well as the more general existence problem for TSQS(\(s^ut^1\)) remains open.