Nonexistence of sparse triple systems over abelian groups and involutions

作者：Fujiwara Yuichiro^*

来源：Journal of Algebraic Combinatorics, 2007, 26(4): 495-506.

DOI：10.1007/s10801-007-0068-1

• Steiner triple system
• r-sparse
• automorphism

In 1973 Paul Erdos 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 >= N13 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.

• 出版日期2007-12