The spectrum for large sets of pure Mendelsohn triple systems

The existence spectrum for overlarge sets of pure directed triple systems

“…For i = 3, 5, there exist a 1-PPDCS(3 i : 1) and a PDF(3 i+1 ) from [8], so there exists a 1-PPDCS(6 k : 4) by Theorem 2.3. Since there exist an OLPDTS (9) and an OLPDTS(9, 3) by Lemmas 4.2 and 4.3, we can get an OLPDTS(6k +3) by Theorem 2.1.…”

“…For k = 1, 2, there exists an OLPDTS (9) and an OLPDTS(15) by Lemma 4.2. For k 3, there exists a 2-FG(3, ({3, 5}, {3, 5}, {4, 6}), 2k) of type 2 k for any k 3 from [4].…”

“…Let (X, G, B 1 , B 2 , B T ) be a 2-FG(3, ({3}, {3}, {4, 6}), 2k) of type 2 k−2 4 1 if k ≡ 2 (mod 3) and k 5, or of type 2 k if k ≡ 0, 1 (mod 3) and k 3, which exists in [9]. We construct the desired design…”

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

“…x ∈ B} where G x = {x} × Z 3 . Such a design exists by [9]. Its block set E B can be partitioned into 12 disjoint families E B (x, i), each of which forms a P MG D D(3 3 …”

“…Since there exist a G D D(3, 4, 2 4 ) and a G D D(3, {4, 6}, 2 6 ) by[7,10], a P M F(6 4 ) and a P M F(6 6 ) also exist by[9], there exist a P M F(12 4 ), a P M F(126 ) and a P M F(4 4 ) by Construction 2.7. (2) Since an O L P MT S(4) exists by Lemma 1.1, there exists a P M F(g n ) for any positive integer g by Construction 2.8.…”