The rainbow Turan number of P5

An edge-colored graph F is rainbow if each edge of F has a unique color. The rainbow Turan number ex & lowast;(n, F) of a graph F is the maximum possible number of edges in a properly edge-colored n-vertex graph with no rainbow copy of F. The study of rainbow Turan numbers was introduced by Keevash, Mubayi, Sudakov, and Verstrete in 2007. In this paper we focus on ex & lowast;(n, P5). While several recent papers have investigated rainbow Tur<acute accent>an numbers for -edge paths P-e, exact results have only been obtained for < 5, and P-5 represents one of the smallest cases left open in rainbow Tur<acute accent>an theory. In this paper, we prove that ex & lowast;(n, P5) <= 5n 2 . Combined with a lower-bound construction due to Johnston and Rombach, this result shows that ex & lowast;(n, P5) = 5n /2 when n is divisible by 16, thereby settling the question asymptotically for all n. In addition, this result strengthens the conjecture that ex & lowast;(n, P) = e/2n+O(1) for all >= 3