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Anti-mitre Steiner triple systems. (English) Zbl 0815.05017
A \((k,\ell)\)-configuration in a Steiner triple system \((V,B)\), is a subset of \(\ell\) triples from \(B\) whose union is a \(k\)-element subset of \(V\). The Pasch configuration is the \((6,4)\)-configuration on a set \(\{a,b, c,d, e,f\}\) with triples \(abe\), \(acf\), \(bdf\), \(cde\). The mitre is the \((7,5)\)-configuration on a set \(\{a,b, c,d, e,f, g\}\) with triples \(abe\), \(acf\), \(adg\), \(bcd\), \(efg\). A Steiner triple system (STS) is anti- Pasch (anti-mitre) if it does not contain any Pasch (mitre) configuration. Moreover, an STS is called \(r\)-sparse if every set of \(r+2\) elements carries fewer than \(r\) triples. Every STS is 3-sparse, is 4-sparse if and only if it is anti-Pasch, and 5-sparse if and only if it is both anti-Pasch and anti-mitre.
This paper makes substantial progress toward characterizing those \(v\) for which there exists an anti-mitre STS of order \(v\), and shows that for at least \({9 \over {16}}\) of the admissible values of \(v\) there exists an anti-mitre STS. The paper includes a table summarising small cyclic STS up to order 57, with the number of cyclic STS which are anti-Pasch or anti-mitre, or both (5-sparse). Also cyclic 5-sparse \(\text{STS} (v)\) are given for orders \(v=19\) and (\(v\equiv 1\) or \(3\pmod 6\)), \(33\leq v\leq 97\). This leads to the conjecture made in the paper that a 5-sparse \(\text{STS} (v)\) exists for all \(v\equiv 1, 3\pmod 6\), \(v\geq 33\). Existence of 5-sparse STS for orders 21, 25, 27 and 31 remains open, while not a single example of a 6-sparse \(\text{STS} (v)\) is currently known.

MSC:
05B07 Triple systems
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