Transitive Kirkman Triple Systems

The existence of a large set of pairwise disjoint Kirkman triple systems ( LKTS ) is an old and interesting combinatorial problem. It came from " Sylvester's problem of the 15 schoolgirls " in 1850. It is a pity that despite considerable effort by many mathematicians, until now the research on LKTS has not advanced very far. The known results and constructions of LKTS(u)s are very limited. Most of them were given by Denniston (see [4, 5, 7]), especially the Denniston's tripling construction of LKTS given in 1979(see [5]), which makes great advancement in this problem.The transitive Kirkman triple systems, which is denoted by TKTS, occupies the central part on the tripling construction. In 1996, a further study about the existence of TKTS has been done by J. Lei (see [12]). He summed up the problem to the existence of a transitive Kirkman triple system of order 3p, where p = 5 (mod 6) is a prime. Unfortunately, not many transitive Kirkman triple systems with such orders are known (see [5, 17]): There exists a transitive Kirkman triple system of order 3p, for p {5,11,17,41}.In this thesis, we use a direct construction to prove that there exists a transitive Kirkman triple system of order 3p, for p e {23, 29, 47, 53, 59, 71, 83, 89, 101, 107, 113, 131, 137}. Further, we show that there always exists a (p, 3,1)- near difference family (NDF) for p = 5(mod 6) is a prime. This kind of combinatorial construction plays a key role on the direct construction for TKTS.This thesis is divided into 4 sections. In section 1, we introduce some correlative background and our main research contents and results; In section 2, some most important conceptions and theorems are presented to build a good foundation for later sections; In section 3, we give some properties and a direct construction of TKTS, moreover, we design programmes and obtain some new TKTS(3p)s with the aid of computer; In section 4, we make further study about this construction theoretically and prove the existence of a related combinatorial construction - the (p, 3,1)- NDF.