The Intersection Numbers Of KTSs With A Common Parallel Class

The problem of intersection numbers is very important in the theory of combina-torial designs and is widely used in statistics.In[14], Rosa first introduced the problem of intersection numbers of KTS in 1980. Later, in 1999, Chang and Lo Faro[2] determined the pairs (k, v) for which there exists a pair of Kirkman triple systems on the same v-set having k triples in common with only 10 pairs of (k, v) undecided. In 2003, Chang and Lo Faro[3] determined the pairs (k,2r+1) for which there exists a pair of Kirkman triple systems on the same (2r+1)-set having k+r triples in common and r of them being the triples of a common flower with only 9 pairs of (k,2r+1) undecided.In this thesis, the intersection numbers of KTSs with a common parallel class are investigated. The notation will be described in Chapter 1. There are four chapters in this thesis:In Chapter 1, we introduce the background of intersection numbers of KTSs and the concepts involved in this paper. We also introduce the relationship between KTS and RGDD. Some auxiliary designs and some methods of recursive constructions are given in this chapter.In Chapter 2, when u=3,5,7, the problem of intersection numbers of KTS(3u) with a common parallel class is completely solved.In Chapter 3, we will introduce some working lemmas and theorems in recursive constructions. When u is an odd integer and u(?)9, the problem of intersection numbers of KTS(3u) with a common parallel class is solved except possibly 23 cases.In Chapter 4, the main results of this thesis are summarized and the further research problem is presented.