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In this thesis we consider the maximum number of points in $mathbb{R}^d$ which form exactly $t$ distinct triangles, which we denote by $F_d(t)$. We determine the values of $F_d(1)$ for all $dgeq3$, as well as determining $F_3(2)$. It was known from the work of Epstein et al. cite{Epstein} that $F_2(1) = 4$. Here we show somewhat surprisingly that $F_3(1) = 4$ and $F_d(1) = d + 1$, whenever $d geq 3$, and characterize the optimal point configurations. We also show that $F_3(2) = 6$ and give one such optimal point configuration. This is a higher dimensional extension of a variant of the distinct distance problem put forward by ErdH{o}s and Fishburn cite{ErdosFishburn}. / Master of Science / In this thesis we consider the following question: Given a number of triangles, t, where each of these triangles are different, we ask what is the maximum number of points that can be placed in d-dimensional space, such that every triplets of these points form the vertices of only the t allowable triangles. We answer this for every dimension, d when the number of triangles is t = 1, as well as show that when t = 2 triangle are in d = 3-dimensional space. This set of questions rises from considering the work of Erd˝os and Fishburn in higher dimensional space [EF].

An MSc dissertation by Shivani Singh. University of Witwatersrand Faculty of Science, School of Mathematics. August 2016. / These symmetries have interesting applications to enumerative combinatorics and to Ramsey theory. The aim of this thesis will be to present some important results in these fields. In particular, we shall enumerate the r-ary symmetric bracelets of length n. / LG2017

Building on the work of Longuet-Higgins in 1972 and Calderbank and Macpherson in 2009, we study the combinatorics of symmetric configurations of hyperplanes and points in projective space, called generalized Cox configurations. To do so, we use the formalism of morphisms between incidence systems. We notice that the combinatorics of Cox configurations are closely related to incidence systems associated to certain Coxeter groups. Furthermore, the incidence geometry of projective space P (V ), where V is a vector space, can be viewed as an incidence system of maximal parabolic subalgebras in a semisimple Lie algebra g, in the special case g = pgl (V ) the projective general linear Lie algebra of V . Using Lie theory, the Coxeter incidence system for the Coxeter group, whose Coxeter diagram is the underlying diagram of the Dynkin diagram of the g, can be embedded into the parabolic incidence system for g. This embedding gives a symmetric geometric configuration which we call a standard parabolic configuration of g. In order to construct a generalized Cox configuration, we project a standard parabolic configuration of type Dn into the parabolic incidence system of projective space using a process called parabolic projection, which maps a parabolic subalgebra of the Lie algebra to a parabolic subalgebra of a lower dimensional Lie algebra. As a consequence of this construction, we obtain Cox configurations and their analogues in higher dimensional projective spaces. We conjecture that the generalized Cox configurations we construct using parabolic projection are nondegenerate and, furthermore, any non-degenerate Cox configuration is obtained in this way. This conjecture yields a formula for the dimension of the space of non-degenerate generalized Cox configurations of a fixed type, which enables us to develop a recursive construction for them. This construction is closely related to Longuet-Higgins’ recursive construction of (generalized) Clifford configurations but our examples are more general and involve the extra parameters.

Thesis (Ph. D.)--University of Rhode Island, 2005. / Typescript. Includes bibliographical references (leaves 73-74).

In this dissertation, we examine the nature of embeddings with regard to both combinatorial and geometric configurations. A combinatorial [r,k]-configuration is a collection of abstract points and sets (referred to as blocks) such that each point is a member of r blocks, each block is of size k, and these objects satisfy a linearity criterion: no two blocks intersect in more than one point. A geometric configuration requires that the points and blocks be realized as points and lines within the Euclidean plane. We provide improvements on the current bounds for the asymptotic existence of both combinatorial and geometric configurations. In addition, we examine the largely new problem of embedding configurations within larger configurations possessing regularity properties. Additionally, previously undiscovered geometric [r,k]-configurations are found as near-coverings of combinatorial configurations. / Graduate

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Thesis (Ph.D.)--University of Adelaide, School of Architecture, Landscape Architecture and Urban Design, 2000? / Bibliography: leaves 231-239.