Rainbow Turan Methods for Trees

The rainbow Turan number, a natural extension of the well-studied traditionalTuran number, was introduced in 2007 by Keevash, Mubayi, Sudakov and Verstraete. The rainbow Tur ́an number of a graph F , ex*(n, F ), is the largest number of edges for an n vertex graph G that can be properly edge colored with no rainbow F subgraph. Chapter 1 of this dissertation gives relevant definitions and a brief history of extremal graph theory. Chapter 2 defines k-unique colorings and the related k-unique Turan number and provides preliminary results on this new variant. In Chapter 3, we explore the reduction method for finding upper bounds on rainbow Turan numbers and use this to inform results for the rainbow Turan numbers of specific families of trees. These results are used in Chapter 4 to prove that the rainbow Turan numbers of all trees are linear in n, which correlates to a well-known property of the traditional Turan numbers of trees. We discuss improvements to the constant term in Chapters 4 and 5, and conclude with a discussion on avenues for future work