  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 $r\le 9$.
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 the fifth
order Tur\'an number for $P$, for all $n$. Using this number for $n=16$, we
confirm the formula $R(P;10)=16$.
