Equitable embeddings of Steiner triple systems
An \(n\)-colouring of a Steiner triple system \(\text{STS}(v)\) is a map which assigns one colour, from a set of \(n\) given colours, to each point in such a way that no triple is monochromatic. An STS is \(r\)-chromatic if it can be \(r\)-coloured but not \((r- 1)\)-coloured. A colouring of an STS is equitable if the cardinalities of the colour classes differ by at most one. The authors prove that if there exists an \(r\)-chromatic \(\text{STS}(w)\) which admits an equitable \(r\)-colouring, then an \(r\)-chromatic \(\text{STS}(v)\) exists which admits an equitable \(r\)-colouring for every admissible \(v\geq 2w+ 1\) when \(w\equiv 3\pmod 6\) and \(v\geq 6w+ 1\) when \(w\equiv 1\pmod 6\). The proof uses an embedding result for pairwise balanced designs, colourings of small systems, and tripling constructions. All known previous results on colourings of STS are briefly surveyed.
scientific article; zbMATH DE number 842849
scientific article