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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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