The Ramsey number R(3, t) has order of magnitude t2/log t
Random Structures & Algorithms
Higher-order Erdös: Szekeres theorems
Proceedings of the twenty-eighth annual symposium on Computational geometry
Erdős-Hajnal-type theorems in hypergraphs
Journal of Combinatorial Theory Series B
Ramsey-type results for semi-algebraic relations
Proceedings of the twenty-ninth annual symposium on Computational geometry
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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 N@?-(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 N@?-(n)"@?^k, then 2^N@?-(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 N@?-(n)"@?^k, then 2^N@?-(n+3)"@?^k^+^1. If also k is odd or @?=3, then we get the better bound 2^N@?-(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.