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Some results on quadrilaterals in Steiner triple systems. (English) Zbl 0783.05022
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.

MSC:
05B07 Triple systems
05B25 Combinatorial aspects of finite geometries
51E10 Steiner systems in finite geometry
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