Triple metamorphosis of twofold triple systems

In a simple twofold triple system ( X , B ) , any two distinct triples T 1 , T 2 with | T 1 ∩ T 2 | = 2 form a matched pair. Let F be a pairing of the triples of B into matched pairs (if possible). Let D be the collection of double edges belonging to the matched pairs in F , and let F ∗ be the collection of 4 -cycles obtained by removing the double edges from the matched pairs in F . If the edges belonging to D can be assembled into a collection of 4 -cycles D ∗ , then ( X , F ∗ ∪ D ∗ ) is a twofold 4 -cycle system called a metamorphosis of the twofold triple system ( X , B ) . Previous work (Gionfriddo and Lindner, 2003  [7]) has shown that the spectrum for twofold triple systems having a metamorphosis into a twofold 4 -cycle system is precisely the set of all n ≡ 0 , 1 , 4 or 9 ( m o d 12 ) , n ≥ 9 . In this paper, we extend this result as follows. We construct for each n ≡ 0 , 1 , 4 or 9 ( m o d 12 ) , n ≠ 9 or 12 , a twofold triple system ( X , B ) with the property that the triples in B can be arranged into three sets of matched pairs F 1 , F 2 , F 3 having metamorphoses into twofold 4 -cycle systems ( X , F 1 ∗ ∪ D 1 ∗ ) , ( X , F 2 ∗ ∪ D 2 ∗ ) , and ( X , F 3 ∗ ∪ D 3 ∗ ) , respectively, with the property that D 1 ∪ D 2 ∪ D 3 = 2 K n . In this case we say that ( X , B ) has a triple metamorphosis. Such a twofold triple system does not exist for n = 9 , and its existence for n = 12 remains an open and apparently a very difficult problem.