Multicolour Ramsey numbers of paths and even cycles
arXiv:1606.00762 · doi:10.1016/j.ejc.2017.03.002
Abstract
We prove new upper bounds on the multicolour Ramsey numbers of paths and even cycles. It is well known that $(k-1)n+o(n)\leq R_k(P_n)\leq R_k(C_n)\leq kn+o(n)$. The upper bound was recently improved by Sárközy who showed that $R_k(C_n)\leq\left(k-\frac{k}{16k^3+1}\right)n+o(n)$. Here we show $R_k(C_n) \leq (k-\frac14)n +o(n)$, obtaining the first improvement to the coefficient of the linear term by an absolute constant.
12 pages, minor changes to exposition