Support sizes of triple systems (Q1199994)

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 \(\text{SS}(v,\lambda)\) of possible support sizes in a \(\text{TS}(v,\lambda)\) is determined for all \(v\) and all \(\lambda\), except that it reamins undecided whether (1) \(s\in\text{SS}(14,6)\) for \(s=45,46,47,48\), and (2) \(s_ v+8\in\text{SS}(v,\lambda)\) when \(v\equiv 8\pmod{12}\), where \(s_ v=\lfloor v(v+2)/6\rfloor\).