Multicolour Ramsey numbers of paths and even cycles

Ewan Davies; Matthew Jenssen; Barnaby Roberts

doi:10.1016/j.ejc.2017.03.002

paper

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 $(k - 1) n + o (n) \leq R_{k} (P_{n}) \leq R_{k} (C_{n}) \leq k n + o (n)$ . The upper bound was recently improved by Sárközy who showed that $R_{k} (C_{n}) \leq (k - \frac{k}{16 k ^{3} + 1}) n + o (n)$ . Here we show $R_{k} (C_{n}) \leq (k - \frac{1}{4}) n + o (n)$ , obtaining the first improvement to the coefficient of the linear term by an absolute constant.

12 pages, minor changes to exposition

Multicolour Ramsey numbers of paths and even cycles

Abstract

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