Simple signed Steiner triple systems. (English) Zbl 1253.05042
Summary: Let \(X\) be a \(v\)-set, \(\mathcal B\) be a set of 3-subsets (triples) of \(X\), and \(\mathcal B^{+}\cup \mathcal B^{-}\) be a partition of \(\mathcal B\) with \(|\mathcal B^{-}|=s\). The pair \((X, \mathcal B)\) 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 \(B\in \mathcal B^{+}\) is one more than the number of occurrences in triples \(B\in \mathcal B^{-}\).
In this paper, we prove that ST\((v,s)\) exists if and only if \(v\equiv 1,3 \pmod 6\), \(v\neq 7\) and \(s \in \{0,1,\dots,s_{v}-6,s_{v}-4,s_v\}\), where \(s_{v} = v(v-1)(v-3)/12\) and for \(v=7\), \(s\in \{0,2,3,5,6,8,14\}\).
In this paper, we prove that ST\((v,s)\) exists if and only if \(v\equiv 1,3 \pmod 6\), \(v\neq 7\) and \(s \in \{0,1,\dots,s_{v}-6,s_{v}-4,s_v\}\), where \(s_{v} = v(v-1)(v-3)/12\) and for \(v=7\), \(s\in \{0,2,3,5,6,8,14\}\).
MSC:
| 05B07 | Triple systems |
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