Refined Turán numbers and Ramsey numbers for the loose 3-uniform path of length three

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\le 7$. The proof of this result relies on a careful analysis of the Tur\'an numbers for $P$. In this paper, we refine this analysis further and compute, for all $n$, the third and fourth order Tur\'an numbers for $P$. With the help of the former, we confirm the formula $R(P;r)=r+6$ for $r\in\{8,9\}$.