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.
@article{bwmeta1.element.doi-10_7151_dmgt_1940,
author = {Joanna Polcyn},
title = {One More Tur\'an Number and Ramsey Number for the Loose 3-Uniform Path of Length Three},
journal = {Discussiones Mathematicae Graph Theory},
volume = {37},
year = {2017},
pages = {443-464},
zbl = {06705139},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1940}
}
Joanna Polcyn. One More Turán Number and Ramsey Number for the Loose 3-Uniform Path of Length Three. Discussiones Mathematicae Graph Theory, Tome 37 (2017) pp. 443-464. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1940/