Triple metamorphosis of twofold triple systems

In a simple twofold triple system (X, B), any two distinct triples T-1, T-2 with vertical bar T-1 boolean AND T-2 vertical bar = 2 form a matched pair. Let F be a pairing of the triples of 2 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* boolean OR 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 equivalent to 0, 1,4 or 9 (mod 12), n >= 9. In this paper, we extend this result as follows. We construct for each n equivalent to 0, 1,4 or 9 (mod 12), n not equal 9 or 12, a twofold triple system (X, B) with the property that the triples in 2 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* boolean OR D-1*). (X, F-2* boolean OR D-2*), and (X, F-3* boolean OR D-3*), respectively, with the property that D-1 boolean OR D-2 boolean OR D-3 = 2K(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. (C) 2011 Elsevier B.V. All rights reserved.