Large sets of extended directed triple systems with odd orders
This paper is concerned with the problem of the existence of large sets of extended directed triple systems with odd orders. Let \(X\) be a finite set with \(v\) elements, and let \(x\), \(y\), \(x\) \((x\neq y\neq z)\) be elements of \(X\). A triple \([x,y,z]\) or (cyclic triple \(\langle x,y,z\rangle\), or transitive triple \((x,y,z)\)) on \(X\) is a set of three unordered pairs \([x,y]\), \([y,z]\), \([z,x]\) (or ordered pairs \((x,y)\), \((y,z)\), \((z,x)\), or ordered pairs \((x,y)\), \((y,z)\), \((x,z)\)) of \(X\). For each of these three types of triples, the corresponding extended triples and their large sets are introduced. An extended Steiner (or Mendelshohn, or directed) triple systems (denoted by \(\text{EMTS}(v)\), or \(\text{EDTS}(v)\)) is a pair \((X,A)\), \(A\) being a collection of extended triples (or cyclic triples, or transitive triples) on \(X\), called blocks, such that each extended unordered (or ordered) pair of \(X\) is contained in exactly one block of \(A\). Corresponding to each extended system, its large set (denoted by \(\text{LEMTS}(v)\), or \(\text{LEMTS}(v)\), or \(\text{LEDTS}(v)\)) is introduced which is a collection \(\langle(X, A_k)\rangle_k\), where \(A_k\) form a partition of all extended triples on \(X\) and each other \((X,A_k)\) is an \(\text{ESTS}(v)\), or \(\text{EMTS}(v)\), or \(\text{EDTS}(v)\). The authors here discuss the existence problem of \(\text{LEDTS}(v)\) when \(v\) is odd. The following are the two main results: Result 1. There exists an \(\text{LEDTS}(v)\) for any odd \(v\) except possible \(v=95\), \(143\), \(167\), \(203\), \(215\). Result 2. There exists an \(\text{LEDTS}(v)\) for any integer \(v\neq 4\) except possible \(v=95\), \(143\), \(167\), \(203\), \(215\).
scientific article; zbMATH DE number 5817555