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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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References:
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