ONE MORE TURAN NUMBER AND RAMSEY NUMBER 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 <= 9. The proof of this result relies on a careful analysis of the Turan numbers for P. In this paper, we refine this analysis further and compute the fifth order Turan number for P, for all n. Using this number for n = 16, we confirm the formula R(P;10) = 16.