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On 6-sparse Steiner triple systems. (English) Zbl 1121.05015
A Steiner triple system of order $$v$$ $$[\text{STS}(v)]$$ consists of a $$v$$-set $$V$$ of elements and a family $$B$$ of 3-subsets of $$V$$ called triples such that each 2-subset of $$V$$ is contained in exactly one triple of $$B$$. A configuration in an $$\text{STS}(v)$$ is a partial triple system consisting typically of a small number of triples. An $$\text{STS}(v)$$ is $$k$$-sparse if it contains no configuration with $$n$$ triples and $$n+2$$ points for any $$4\leq n\leq k$$. The 4-sparse STSs are precisely those which are anti-Pasch, and are known to exist for all orders $$v$$ congruent to 1 or 3 modulo 6, except for $$v= 7$$ or 13. The 5-sparse STSs are precisely those which are both anti-Pasch and anti-mitre, and have been shown by A. Wolfe [Electron. J. Comb. 12, No. 1, Research paper R68, 42 p., electronic only (2005; Zbl 1079.05013)] to exist for almost all admissible orders.
In this article, the authors use a construction of M. J. Grannell, T. S. Griggs and J. P. Murphy [J. Comb. Des. 7, No. 5, 327–330 (1999; Zbl 0935.05016)] to obtain first examples of 6-sparse STSs. Their 29 examples given range from $$v= 139$$ to $$v= 4447$$; it is shown that these are the only 6-sparse STSs that can be obtained by using this method. A tripling construction and a product construction are then used to show that there exist infinitely many 6-sparse STSs. Also in this paper, the authors construct a new perfect STS of order $$v= 135$$,859 and a new uniform (non-perfect) STS with $$v= 180$$,907.
Reviewer: Colin Reid

MSC:
 05B07 Triple systems
Full Text:
References:
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