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Abstract

This dissertation is comprised of five chapters, each primarily concerning a problem in Euclidean geometry. In Chapter 2 we discuss bounds on axial and folding symmetry in convex bodies. We show new upper and lower bounds for both notions of symmetry in two dimensions improving results of Choi and Lassak, and give some upper bounds in higher dimensions. In Chapter 3 we obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid. In Chapter 4, a procedure for generating dense unit distance graphs in two and three dimensions computationally is given. It begins with an approximate algorithm based on 'gravity', which is combined with some exact techniques. We use this algorithm to find graphs that match the known and suspected optimal densities. In Chapter 5, we introduce the notation 𝔼ⁿ → (ℓᵣ,ℓ𝓈), which is the statement "any red-blue colouring of 𝔼ⁿ has a red congruent copy of ℓᵣ or a blue ℓ𝓈", where ℓ𝓂 denotes a collection of 𝓂 collinear points separated by unit distances. We show here that 𝔼² → (ℓ₃,ℓ₃), which verifies a natural case of a conjecture of Graham. In Chapter 6, we provide a general framework to construct spherical colorings avoiding short monochromatic arithmetic progressions. We show 𝔼ⁿ ↛ (ℓ₃,ℓ₂₀), improving the best-known result 𝔼ⁿ ↛ (ℓ₃,ℓ₁₁₇₇) by Führer and Tóth, and also establish 𝔼ⁿ ↛ (ℓ₄,ℓ₁₄) and E^n ↛ (ℓ₅,ℓ₈) in the spirit of the classical result 𝔼ⁿ ↛ (ℓ₆,ℓ₆) due to Erdős et al. We also show a number of similar three-coloring results, as well as 𝔼ⁿ ↛ (ℓ₃, αℓ₆₈₈₉), where α is an arbitrary positive number scaling the ℓ₆₈₈₉ progression.