Refined Tur\'an numbers and Ramsey numbers for the loose 3-uniform path of length three
Abstract
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 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∈\8,9\.
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