Multicolour Ramsey numbers of paths and even cycles
Ewan Davies, Matthew Jenssen, Barnaby Roberts

TL;DR
This paper improves the upper bounds on multicolour Ramsey numbers for paths and even cycles, notably reducing the coefficient of the linear term in the bounds for cycles.
Contribution
It presents a new upper bound for the multicolour Ramsey numbers of even cycles, improving the linear coefficient by an absolute constant.
Findings
New upper bound: R_k(C_n) ≤ (k - 1/4)n + o(n)
First improvement to the linear coefficient by an absolute constant
Advances understanding of multicolour Ramsey numbers for cycles
Abstract
We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that . The upper bound was recently improved by S\'ark\"ozy who showed that . Here we show , obtaining the first improvement to the coefficient of the linear term by an absolute constant.
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