An extended Mendelsohn triple system of order \(v\) (EMTS(\(v\))) is a collection of cyclically ordered triples of the type \([x,y,z], [x,x,y]\), or \([x,x,x]\) chosen from a \(v\)-set, such that each ordered pair (not necessarily distinct) belongs to exactly one triple. If such a design with parameters \(v\) and \(a\) exist, then they will have \(b_{v,a}\) blocks, where \(b_{v,a} = (v^2 + 2a)/3\). In this paper, we show that there are two (not necessarily distinct) EMTS(\(v\))’s with common triples in the following sets:
\(\{0,1,2,\ldots,b_v-4,b_v-2,b_v\}\), if \(v \neq 6\); and
\(\{0,1,2,\ldots,b_v-4,b_v-2\}\), if \(v = 6\),
where \(b_v\) is \(b_{v,v-1}\) if \(v \equiv 2 \pmod{3}\); \(b_{v,v}\) if \(v \not\equiv 2 \pmod{3}\).
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