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].