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 m1, m2, . . . , ms with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${t = \sum_{i=1}^s m_i}$$\end{document} , does there exist a partial Steiner triple system with t triples whose triples can be partitioned into partial parallel classes of sizes m1, . . . , ms? An affirmative answer to the first question gives an affirmative answer to the second when mi ≤ m for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i \in \{1,2,\ldots,s\}}$$\end{document} . 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\frac{1}{3}\left(v-(9v)^{2/3}\right)+{O}\left(v^{1/3}\right)}$$\end{document} .