Support sizes of sixfold triple systems (Q1801692)

The support size of a triple system \(\text{TS}(v,\lambda)\) of order \(v\) and index \(\lambda\) is the number of distinct triples. The set \(\text{SS}(v,6)\) of possible support sizes in a \(\text{TS}(v,6)\) is almost completely determined. It remains undecided whether \(s_ v+8\in\text{SS}(v,6)\) when \(v\equiv 8\pmod{12}\), where \(s_ v=\lfloor v(v+2)/6\rfloor\). The authors also establish necessary conditions for the determination of possible support sizes of triple systems \(\text{TS}(v,\lambda)\) for any \(\lambda>0\).