Trails of triples in partial triple systems

Given v, t, and m, does there exist a partial Steiner triple system of order v with t triples whose triples can be ordered so that any m consecutive triples are pairwise disjoint? Given v, t, and m ₁,m ₂,.. ,m _s with t = Σ _i=1 ^s m _i, does there exist a partial Steiner triple system with t triples whose triples can be partitioned into partial parallel classes of sizes m ₁,.. ,m _s? An affirmative answer to the first question gives an affirmative answer to the second when m _i ≥ m for each i ε {1, 2,.. ., s}. These questions arise in the analysis of erasure codes for disk arrays and that of codes for unipolar communication, respectively. A complete solution for the first problem is given when m is at most 1/3 (v - (9v) ^2/3)+O(v ^1/3).