On Ramsey numbers of hedgehogs

02/26/2019

∙

by   Jacob Fox, et al.

∙

0

∙

share

The hedgehog H_t is a 3-uniform hypergraph on vertices 1,...,t+t2 such that, for any pair (i,j) with 1< i<j< t, there exists a unique vertex k>t such that {i,j,k} is an edge. Conlon, Fox, and Rödl proved that the two-color Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-color Ramsey number grows exponentially in the number of its vertices. They asked whether the two-color Ramsey number of the hedgehog H_t is nearly linear in the number of its vertices. We answer this question affirmatively, proving that r(H_t) = O(t^2 t).