Paths of Length Three are Kr+1-Turan-Good

The generalized Tur ' an problem ex(n, T, F) is to determine the maximal number of copies of a graph T that can exist in an F-free graph on n vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Tur ' an problem is often the original Tur ' an graph. They gave the name "F-Tur ' an-good" to graphs T for which, for large enough n, the solution to the generalized Tur ' an problem is realized by a Tur ' an graph. They prove that the path graph on two edges, P2, is Kr+1-Tur ' an-good for all r 3, but they conjecture that the same result should hold for all P`. In this paper, using arguments based in flag algebras, we prove that the path on three edges, P3, is also Kr+1-Tur ' an-good for all r 3.