Partitions of sets of two-fold triple systems, and their relations to some strongly regular graphs
A two-fold triple system (TTS) is a 2-\((v,k,2)\) design (and hence \(v \equiv 1\) or \(3 \pmod 6\)). An overlarge set of TTS, denoted by \(\text{OS(TTS} (v))\) is a set of \(v + 1\) mutually disjoint \(\text{TTS} (v)\) (i.e. they have no triple in common), each of them based on a different \(v\)-subset of a set \(X\) of cardinality \(v + 1\). The authors prove that if \(|X |= 7\), then the 84 distinct (but all isomorphic) TTS(6) based on \(X\), can be partitioned into 12 OS(TTS(6)) each containing 7 TTS(6). Moreover there exists a resolution into 84 parallel classes of the set of 1008 distinct OS(TTS(6)) based on \(X\). Using this terminology they revisit the Hoffman-Singleton graph as well as the Higman-Sims graph. They proof, using their terminology, the well-known fact that this last graph is the union of two Hoffman-Singleton graphs. The reviewer disagrees with the authors that their description of the Higman-Sims graph is the first elementary construction.
scientific article; zbMATH DE number 837841
scientific article