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

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