  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 $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$,
and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a
standard application of the Tur\'an number $ex_3(n;P)$, which was determined to
be $\binom{n-1}2$ in our previous work. We have also shown that the full star
is the only extremal 3-graph for $P$. In this paper, we perform a subtle
analysis of the Tur\'an numbers for $P$ under some additional restrictions.
Most importantly, we determine the largest number of edges in an $n$-vertex
$P$-free 3-graph which is not a star. These Tur\'an type results, in turn,
allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.
