Hypergraph packing and sparse bipartite ramsey numbers

Authors:
David Conlon
Affiliations:
St john's college, cambridge cb2 1tp, uk (e-mail: d.conlon@dpmms.cam.ac.uk)
Venue:
Combinatorics, Probability and Computing
Year:
2009

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Abstract

We prove that there exists a constant c such that, for any integer Δ, the Ramsey number of a bipartite graph on n vertices with maximum degree Δ is less than 2cΔn. A probabilistic argument due to Graham, Rödl and Ruciński implies that this result is essentially sharp, up to the constant c in the exponent. Our proof hinges upon a quantitative form of a hypergraph packing result of Rödl, Ruciński and Taraz.