Paths of Length Three are $K_{r+1}$-Tur\'an Good

The generalized Tur\'an problem $ext(n,T,F)$ 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, $P_2$, is $K_{r+1}$-Tur\'an-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Tur\'an-good for all $r \ge 3$.