New lower bounds for hypergraph Ramsey numbers

The Ramsey number rk(s, n) is the minimum N such that for every red-blue coloring of the k-tuples of {1, . . . , N}, there are s integers such that every k-tuple among them is red, or n integers such that every k-tuple among them is blue. We prove the following new lower bounds for 4-uniform hypergraph Ramsey numbers: r4(5, n) > 2 n log n and r4(6, n) > 2 2 1/5 , where c is an absolute positive constant. This substantially improves the previous best bounds of 2 c log log n and 2 c log n , respectively. Using previously known upper bounds, our result implies that the growth rate of r4(6, n) is double exponential in a power of n. As a consequence, we obtain similar bounds for the k-uniform Ramsey numbers rk(k+ 1, n) and rk(k + 2, n) where the exponent is replaced by an appropriate tower function. This almost solves the question of determining the tower growth rate for all classical off-diagonal hypergraph Ramsey numbers, a question first posed by Erdős and Hajnal in 1972. The only problem that remains is to prove that r4(5, n) is double exponential in a power of n.