Abstract
The Ramsey number r k ( 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: r 4 ( 5 , n ) > 2 n c log n and r 4 ( 6 , n ) > 2 2 c n 1 / 5 , where c is an absolute positive constant. This substantially improves the previous best bounds of 2 n c log log n and 2 n c log n , respectively. Using previously known upper bounds, our result implies that the growth rate of r 4 ( 6 , n ) is double exponential in a power of n. As a consequence, we obtain similar bounds for the k-uniform Ramsey numbers r k ( k + 1 , n ) and r k ( 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 r 4 ( 5 , n ) is double exponential in a power of n.
Submitted Version (
Free)
Published Version
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call