Nonexistence of sparse triple systems over abelian groups and involutions |
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Authors: | Yuichiro Fujiwara |
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Institution: | (1) Graduate School of Information Science, Nagoya University, Furo-cho, Chikusa-ku 464-8601, Japan |
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Abstract: | In 1973 Paul Erdős conjectured that there is an integer v
0(r) such that, for every v>v
0(r) and v≡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≤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≥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.
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Keywords: | Steiner triple system r-Sparse Automorphism |
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