Geodesic
Nonexistence of sparse triple systems over abelian groups and involutions
Journal of Algebraic Combinatorics, Tome 26 (2007) no. 4, pp. 495-506.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: In 1973 Paul Erdős conjectured that there is an integer $v _{0}( r)$ such that, for every $v> v _{0}( r)$ and $v\equiv 1$,3 (mod 6), there exists a Steiner triple system of order $v$, containing no $i$ blocks on $i+2$ points for every $1 i\leq r$. Such an STS is said to be $r$-sparse. In this paper we consider relations of automorphisms of an STS to its sparseness. We show that for every $r\geq 13$ there exists no point-transitive $r$-sparse STS over an abelian group. This bound and the classification of transitive groups give further nonexistence results on block-transitive, flag-transitive, 2-transitive, and 2-homogeneous STSs with high sparseness. We also give stronger bounds on the sparseness of STSs having some particular automorphisms with small groups. As a corollary of these results, it is shown that various well-known automorphisms, such as cyclic, 1-rotational over arbitrary groups, and involutions, prevent an STS from being high-sparse.
Classification : 05B07, 20B25
Keywords: keywords Steiner triple system, $r$-sparse, automorphism 
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     author = {Fujiwara, Yuichiro},
     title = {Nonexistence of sparse triple systems over abelian groups and involutions},
     journal = {Journal of Algebraic Combinatorics},
     pages = {495--506},
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     year = {2007},
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     url = {http://geodesic.mathdoc.fr/item/JAC_2007__26_4_a2/}
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Fujiwara, Yuichiro. Nonexistence of sparse triple systems over abelian groups and involutions. Journal of Algebraic Combinatorics, Tome 26 (2007) no. 4, pp. 495-506. http://geodesic.mathdoc.fr/item/JAC_2007__26_4_a2/