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Ramsey-type numbers involving graphs and hypergraphs with large girth
Published online by Cambridge University Press: 12 April 2021
Abstract
Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle Ck. The existence of such graphs H was confirmed by the third author and Ruciński.
We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(Ck; r) is the r-colour Ramsey number for the cycle Ck) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic Ck Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.
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- © The Author(s), 2021. Published by Cambridge University Press
Footnotes
H. Hàn was partly supported by FAPESP (2010/16526-3 and 2013/11353-1).
V. Rödl was supported by NSF grants DMS 1301698 and 1764385.
M. Schacht was supported through the Heisenberg-Programme of the DFG.