On Large Set Of Kirkman Triple Systems

In 1850, T. P. Kirkman posed the following problem: Fifteen young ladies in a school walk out three abreast for seven days in succession: it is required to arrange them daily, so that no two will walk twice abreast. In the same year, J. J. Sylvester posed the further problem: Every three girls walk together exactly once in a 13 week period. The problem is just the known so-called Sylvester's problem of the 15 schoolgirls, which was solved by R. H. F. Denniston until 1974."Sylvester's problem of the 15 schoolgirls" is first large set problem in mathematical history. The generalized case of the problem, unlimited number of points is 15, is called "large set of disjoint Kirkman triple systems (LKTS}". Until now the research on LKTS has not advanced very far. By the end of 1979, Denniston had given the direct constructions for several small values and a tripling recursive construction of LKTS. Meanwhile, S. Schreiber had also given the existence of an LKTS(33}. Furthermore, some infinite classes can be gotten by using the Denniston's recursive construction. The research on LKTS has been almost at a standstill since then. Especially, new recursive constructions were not found. In this thesis, we introduce concepts of large sets of generalized Kirkman systems (LGKS) and overlarge sets of Kirkman frames (OLKF), and present some constructions and new results for large set of Kirkman triple systems. Our main results is:(l)There exists an LKTS(v) for (2) If there exist both an OLKF(6k) and an LGKS({3}, then there exists an LKTS(v) for any v = 3 (mod 6), v 21.