An improved bound for the stepping-up lemma

The partition relation N->(n)"@?^k means that whenever the k-tuples of an N-element set are @?-colored, there is a monochromatic set of size n, where a set is called monochromatic if all its k-tuples have the same color. The logical negation of N->(n)"@?^k is written as [email protected]?->(n)"@?^k. An ingenious construction of Erdos and Hajnal known as the stepping-up lemma gives a negative partition relation for higher uniformity from one of lower uniformity, effectively gaining an exponential in each application. Namely, if @?>=2, k>=3, and [email protected]?->(n)"@?^k, then 2^[email protected]?->(2n+k-4)"@?^k^+^1. In this paper we give an improved construction for k>=4. We introduce a general class of colorings which extends the framework of Erdos and Hajnal and can be used to establish negative partition relations. We show that if @?>=2, k>=4 and [email protected]?->(n)"@?^k, then 2^[email protected]?->(n+3)"@?^k^+^1. If also k is odd or @?>=3, then we get the better bound 2^[email protected]?->(n+2)"@?^k^+^1. This improved bound gives a coloring of the k-tuples whose largest monochromatic set is a factor @W(2^k) smaller than that given by the original version of the stepping-up lemma. We give several applications of our result to lower bounds on hypergraph Ramsey numbers. In particular, for fixed @?>=4 we determine up to an absolute constant factor (which is independent of k) the size of the largest guaranteed monochromatic set in an @?-coloring of the k-tuples of an N-set.