A Completion of the Spectrum for the Overlarge Sets of Pure Mendelsohn Triple Systems

Abstract

A pure Mendelsohn triple system of order v, denoted by PMTS(v), is a pair $(X,\mathcal {B})$ where X is a v-set and $\mathcal {B}$ is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of $\mathcal {B}$ and if $\langle a,b,c\rangle \in \mathcal {B}$ implies $\langle c,b,a\rangle \notin \mathcal {B}$. An overlarge set of PMTS(v), denoted by OLPMTS(v), is a collection $\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i$, where Y is a $(v+1)$-set, $y_i\in Y$, each $(Y{\setminus }\{y_i\},{\mathcal {A}}_i)$ is a PMTS(v) and these ${\mathcal {A}}_i$s form a partition of all cyclic triples on Y. It is shown in [3] that there exists an OLPMTS(v) for $v\equiv 1,3$ (mod 6), $v>3$, or $v \equiv 0,4$ (mod 12). In this paper, we shall discuss the existence problem of OLPMTS(v)s for $v\equiv 6,10$ (mod 12) and get the following conclusion: there exists an OLPMTS(v) if and only if $v\equiv 0,1$ (mod 3), $v>3$ and $v\ne 6$.