# Some constructions of combinatorial designs

Ellison, Leigh H. M (2006) Some constructions of combinatorial designs. PhD thesis, University of Glasgow.

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## Abstract

The objects of study of this thesis are combinatorial designs. Chapters 2 and 3 deal with various refinements of whist tournament, while Chapters 3 and 4 focus on terraces. Chapter 2 is devoted to the investigation of Z-cyclic ordered triplewhist tournaments on p elements, where p = 5 (mod 8); Z-cyclic ordered triplewhist and directed triplewhist tournaments on p elements, where p = 9 (mod 16); and Z-cyclic directed moore (2,6) generalised whist tournament designs on p elements, where p = 7 (mod 12). In each of these cases, p is prime. In an effort to prove the existence of an infinite family of each of these tournaments, constructions are introduced and the conditions under which they give the initial round of a tournament of the kind we desire are found. A bound above which these conditions are always satisfied is then obtained, and we try to fill in the appropriate gaps below that bound. In Chapter 3 we investigate the existence of tournaments of the type seen in Chapter 2 which involve four players per game, with an additional property. This is known as the three person property and is defined in Chapter 1. Here, we focus on one of the constructions introduced in Chapter 2 for each type of tournament. Then we find a new bound using only that construction with the additional conditions introduced by the three person property, and again try to fill in the appropriate gaps below the bound. Chapter 4 is an investigation of logarithmic terraces and their properties. Very little work has been done on them previously, so this was really an opportunity to look at them more closely in an effort to find as many interesting properties as possible. Some general results and examples are given, with the focal point of the chapter being the study of terraces which are simultaneously logarithmic for two different primitive roots. In Chapter 5, a more specific problem is addressed which involves training schedules for athletes. Here we want n(n - 1) athletes to carry out n tasks in some order, then keep repeating them in different orders in blocks of n as many times as possible so that certain conditions are satisfied. These conditions are listed in Chapter 5. We make use of the Williams terrace and the Owens terrace in our attempt to find a general method which allows the given conditions to be satisfied and gets as close as possible to the theoretical limit where each athlete carries out the n tasks n - 1 times.

Item Type: Thesis (PhD) Doctoral Adviser: Ian Anderson Mathematics 2006 Enlighten Team glathesis:2006-74081 Copyright of this thesis is held by the author. 23 Sep 2019 15:33 23 Sep 2019 15:33 https://theses.gla.ac.uk/id/eprint/74081