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Further 6-sparse Steiner triple systems. (English) Zbl 1188.05034
Erdös conjectured that for each \(k\geq 4\) there exists \(v(k)\) so that for every admissible \(v>v(k)\) there exists a Steiner Triple System on \(v\) points, \(STS(v),\) which does contains no configuration having \(n\) blocks on \(n+2\) points for \(n\) with \(4\leq n\leq k.\) Such \(STS(v)\) is called to be \(k\)-sparse. The existence of \(4\)-sparse systems for all admissible values of \(v,\) and the existence of \(5\)-sparse systems for almost all admissible values of \( v\) has been established. In this paper the authors construct a \(6\)-sparse \(STS(v)\) for \(v=3p,\) where \(p\) is sufficiently big prime \(p\equiv 3\pmod {4}\).

MSC:
05B07 Triple systems
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