Anti-pasch and pasch steiner triple systems
This thesis concentrates on the study of anti-Pasch and Pasch Steiner triple systems and existence results / concerning these. After the introductory Chapter, Chapters 2, 3, 4, 5 treat anti-Pasch systems and the last the reverse problem of Pasch systems. Chapter 1 introduces the basic concepts and definitions including fundamental results on Steiner systems and a brief survey of current knowledge on existence. Chapter 2 concerns the existence of a large set of anti-Pasch systems S(2, 3, v) for certain values of v. This is based on an old general construction for Steiner triple systems which are used to generate the large set. It is found that v is restricted by a number theoretic condition. This Chapter is contained in the paper by Grannell, Griggs, Phelan [G2.2]. Chapter 3 establishes the existence of anti-Pasch Steiner triple systems on v points for all v = 3 (mod 6). This is acheived via an extension of a construction due to Bose [Bi] and certain particular anti-Pasch systems. This work may be found in Griggs, Murphy, Phelan [G3.1]. Chapter 4 investigates the accompanying problem to that of the previous Chapter, anti-Pasch systems with v = 1 (mod 6). This problem is not completely solved unlike that of Chapter 3 and we must content ourselves with classes of such systems. Such classes were found by Robinson [R2] and via a recursive procedure by Stinson and Wei [S3.3] which are both presented here. In addition an improvement of the latter result is given. Chapter 5 looks at the construction of anti-Pasch systems on a countably infinite base set and after the description of all such Steiner triple systems (not necessarily anti- Pasch) it is proved that there is a large set of anti- Pasch countably infinite systems. This Chapter is contained in the paper by Grannell, Griggs, Phelan [G2.3]. Finally Chapter 6 concerns the opposite problem of the existence of systems which may be resolved into Pasch configurations. After looking at the general Pasch content of triple systems as per Stinson and Wei [53.3] there is established the existence of infinite classes of Pasch-resolvable systems following work done by Griggs, de Resmini, Rosa [G3.2] and Griggs [G3.3].
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