Constructions in Ramsey theory

Abstract

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number r 4 ( 5 , n ) , and the same for the iterated ( k − 4 ) -fold logarithm of the k-uniform version r k ( k + 1 , n ) . This is the first improvement of the original exponential lower bound for r 4 ( 5 , n ) implicit in work of Erdős and Hajnal from 1972 and also improves the current best known bounds for larger k due to the authors. Second, we prove an upper bound for the hypergraph Erdős–Rogers function f k + 1 , k + 2 k ( N ) that is an iterated ( k − 13 ) -fold logarithm in N. This improves the previous upper bounds that were only logarithmic and addresses a question of Dudek and the first author that was reiterated by Conlon, Fox and Sudakov. Third, we generalize the results of Erdős and Hajnal about the 3-uniform Ramsey number of K 4 minus an edge versus a clique to k-uniform hypergraphs.