Resolvable twofold triple systems without repeated triples (Q1813660)

A triple system \(S_ \lambda(2,3,v)\) is an ordered pair \((V,\mathbb{B})\), where \(V\) is a \(v\)-set, \(\mathbb{B}\) is a collection of 3-subsets (called triples) of \(V\) such that each pair of distinct elements of \(V\) is contained in exactly \(\lambda\) triples. An \(S_ 2(2,3,v)\) is called a twofold triple system. A parallel class in an \(S_ \lambda(2,3,v)\) is a set of triples which partitions \(V\). An \(S_ \lambda(2,3,v)\) is called resolvable and denoted by \(RS_ \lambda(2,3,v)\) if the triples can be partitioned into parallel classes. An almost parallel class in an \(S_ \lambda(2,3,v)\) is a set of triples which partitions \(V\backslash\{x\}\) for some \(x\in V\). An \(S_ \lambda(2,3,v)\) is called almost resolvable and denoted by \(ARS_ \lambda(2,3,v)\) if the triples can be partitioned into almost parallel classes. An \(S_ \lambda(2,3,v)\) is called simple if it contains no repeated triples. A simple \(S_ 2(2,3,v)\) is known to exist iff \(v\equiv 0\) or \(1\pmod 3\) and \(v\neq 3\) (cf. \textit{J. van Buggenhaut} [Discrete Math. 8, 105-109 (1974; Zbl 0276.05021)] and \textit{D. R. Stinton} and \textit{W. D. Wallis} [Discrete Math. 47, 125-228 (1983; Zbl 0523.05010)]). It is well known that there exists an \(RS_ 2(2,3,v)\) iff \(v\equiv 0\pmod 3\) and \(v\neq 6\); there exists an \(ARS_ 2(2,3,v)\) if \(v\equiv 1\pmod 3\) (see \textit{H. Hanani} [J. Comb. Theory, Ser. A 17, 275-289 (1974; Zbl 0305.05010)]). But the existence of a simple \(RS_ 2(2,3,v)\) and a simple \(ARS_ 2(2,3,v)\) remained to be established. The purpose of this report is to prove the following Theorem 1: (i) There exists a simple \(ARS_ 2(2,3,v)\) iff \(v\equiv 1\pmod 3\). (ii) There exists a simple \(RS_ 2(2,3,v)\) iff \(v\equiv 0\pmod 3\) and \(v>6\).