Ramsey numbers of cycles versus general graphs

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The Ramsey number R(F, H) is the minimum number N such that any N-vertex graph either contains a copy of F or its complement contains H. Burr in 1981 proved a pleasingly general result that, for any graph H, provided n is sufficiently large, a natural lower bound construction gives the correct Ramsey number involving cycles: R(C-n, H) = (n - 1)( x(H) - 1) + sigma (H), where sigma(H) is the minimum possible size of a colour class in a x(H)-colouring of H. Allen, Brightwell and Skokan conjectured that the same should be true already when n >= |H|x(H).We improve this 40-year-old result of Burr by giving quantitative bounds of the form n >= C|H| log(4 )x (H), which is optimal up to the logarithmic factor. In particular, this proves a strengthening of the Allen-Brightwell-Skokan conjecture for all graphs H with large chromatic number.
Publisher
CAMBRIDGE UNIV PRESS
Issue Date
2023-02
Language
English
Article Type
Article
Citation

FORUM OF MATHEMATICS SIGMA, v.11

ISSN
2050-5094
DOI
10.1017/fms.2023.6
URI
http://hdl.handle.net/10203/305813
Appears in Collection
MA-Journal Papers(저널논문)
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