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