The indecomposable partition problem is to partition the set of all triples on v elements into s indecomposable triple systems, where the i th triple system has index λi and λ1 + … +λs = v–2. A complete solution for v ≤ 10 is given here. Extending a construction of Rosa for large sets, we then give a v → 2v + 1 construction for indecomposable partitions. This recursive construction employs solutions to a related partition problem, called indecomposable near-partition. A partial solution to the indecomposable near-partition problem for v = 10 then establishes that for every order v = 5.2i–1, all indecomposable partitions having λi = 1,2 for each i can be realized.
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