Simple signed Steiner triple systems

Details Simple signed Steiner triple systems Simple signed Steiner triple systems

2011-05-14

arxiv.org/abs/1105.2896v2

Mathematics

Combinatorics

11 pages, 1 table. Final version. To appear in Journal of Combinatorial Designs

Scientific paper

Let $X$ be a $v$-set, $\B$ a set of 3-subsets (triples) of $X$, and $\B^+\cup\B^-$ a partition of $\B$ with $|\B^-|=s$. The pair $(X,\B)$ is called a simple signed Steiner triple system, denoted by ST$(v,s)$, if the number of occurrences of every 2-subset of $X$ in triples $B\in\B^+$ is one more than the number of occurrences in triples $B\in\B^-$. In this paper we prove that $\st(v,s)$ exists if and only if $v\equiv1,3\pmod6$, $v\ne7$, and $s\in\{0,1,...,s_v-6,s_v-4,s_v\}$, where $s_v=v(v-1)(v-3)/12$ and for $v=7$, $s\in\{0,2,3,5,6,8,14\}$.

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