Geodesic
One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three
Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 2, pp. 443-464

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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 Turán numbers for P. In this paper, we refine this analysis further and compute the fifth order Turán number for P, for all n. Using this number for n = 16, we confirm the formula R(P; 10) = 16.
Keywords: Ramsey numbers, Turán numbers 
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Polcyn, Joanna. One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three. Discussiones Mathematicae. Graph Theory, Tome 37 (2017) no. 2, pp. 443-464. http://geodesic.mathdoc.fr/item/DMGT_2017_37_2_a9/