Large Sets And Overlarge Sets Of Triple Systems With Resolvability

In 1847, T. P. Kirkman posed the problem about 15 schoolgirls. After that, J. J. Sylvester posed the famous Sylvester's problem in 1850. The generalized case of this problem is called large set of disjoint Kirkman triple systems LKTS(v), which is the first large set problem on combinatorial designs. But, the research for LKTS(v) is far from its being solved completely, due to its high difficulty.In 1991, M. J. Sharry and A. P. Street introduced the concept—overlarge set and completed the existence spectrum for OLSTS(v), overlarge set of STS(v). Following it, overlarge sets of MTS(v) (DTS(v)) and overlarge sets of some special graph designs have been researched. But up to now, the result about OLKTS(v) is less.In this dissertation, we first give a tripling construction for OLKTS(v). Then, introduce a new concept—OLGKS, generalized overlarge sets of Kirkman triple systems, and show the relationship between the new concept and OLKTS(v). Next, some recursive constructions for OLGKS are given, and some new existence results of OLKTS(v) are obtained. Meanwhile, we give an effective approach which can be used to solve the existence for OLKTS(v) completely. Further, we research the existence of LKTS(6v + 3) via F(3,4, (v + 1)(2)) and OLKTS(2v+ 1) with a special property and give some infinite families of large sets of LKTS(6v + 3). In addition, some new results for the existence of LRMTS(v), LRDTS(v), OLRMTS(v) and OLRDTS(v) are gotten. Final, we discuss overlarge sets of graph designs for the graphs P₃, P₃¹, P₃², P₃³ and T₃, and obtain a series of new results.