The Large Sets And Overlarge Sets Of Three Triple Systems

The large set problem in the combinatorial design theory has a long history and a veryimportant applications in experimental design, coding theory, etc. The progress of the relatedresearch had been quite slow during a long period due to its sophistication. Being motivatedby some new methods and techniques, the research in the large set problem has taken on apromising posture in recent thirty years.Steiner triple system, Mendelsohn triple system, transitive triple system and their largesets have all been extensively studied. For these classical triple systems, the elements in eachtriple must be distinct. When this restriction is broken, i.e., the elements in a triple can bethe same, we have the so-called extended Steiner triple system, extended Mendelsohn triplesystem and extended transitive triple system which were firstly introduced by D. M. Johnsonand N. S. Mendelsohn in 1972. The existence problems of these three kinds of extended triplesystems and the existence spectrums of the large sets of extended Steiner triple systems andextended Mendelsohn triple systems are all solved. In this dissertation, we shall focus ondiscussing the existence spectrum of the large set of extended directed triple systems.In 1991, M. J. Sharry and A. P. Street introduced the concept about overlarge set anddetermined the existence spectrum for OLSTS(v), overlarge set of STS(v). Following it,overlarge sets of MTS(v), overlarge sets of DTS(v) and overlarge sets of some graph de-signs have been researched respectively. In this dissertation, we shall discuss the overlargesets of pure directed triple systems, P₃-decompositions and directed P₃-decompositions anddetermined their existence spectrums.The dissertation is divided into four chapters as follows.In Chapter 1, we introduce some terminologies and basic concepts, list some knownresults about extended triple systems, pure directed triple system, P₃, directed P₃ and theirlarge sets. As well, the main aim and conclusions contained in this dissertation are shown.In Chapter 2, the large set of extended directed triple systems LEDTS(v) is discussedin detail. Finally, we prove that there exists an LEDTS(v) for any positive integer v = 4,except five possible orders v = 95, 143, 167, 203, 215.In Chapter 3, we completely determine the existence spectrum of pure directed triple systems.In Chapter 4, for P₃ and P₃³ , we determine the existence spectrums for the overlarge setsof their graph designs, respectively.