Some results on quadrilaterals in Steiner triple systems (Q1199487)

A quadrilateral in a Steiner triple system is a subset of four blocks whose union has precisely 6 points; that is, a quadrilateral is isomorphic to the 4 blocks \{1,2,3\}, \{1,4,5\}, \{2,5,6\} and \{3,4,6\}. The authors are interested in determining for which values of \(n\) there exists a Steiner triple system on \(n\) point, \(\text{STS}(n)\), with no quadrilateral. Two recursive constructions of such \(\text{STS}(n)\) are presented. Let \(\text{MQ} (n)\) be the maximum number of quadrilaterals possible in an \(\text{STS}(n)\). It is shown that \(\text{MQ}(n) \leq n(n- 1)(n-3)/24\) and that equality is obtained if and only if the \(\text{STS}(n)\) is isomorphic to a projective geometry \(\text{PG}(k,2)\) for some \(k \geq 2\). Lower bounds for \(\text{MQ}(n)\) are also studied.