On the Multi-Colored Ramsey Numbers of Paths and Even Cycles

  title={On the Multi-Colored Ramsey Numbers of Paths and Even Cycles},
  author={G{\'a}bor N. S{\'a}rk{\"o}zy},
  journal={Electron. J. Comb.},
In this paper we improve the upper bound on the multi-color Ramsey numbers of paths and even cycles. More precisely, we prove the following. For every $r\geq 2$ there exists an $n_0=n_0(r)$ such that for $n\geq n_0$ we have $$R_r(P_n) \leq \left( r - \frac{r}{16r^3+1} \right) n.$$ For every $r\geq 2$ and even $n$ we have $$R_r(C_n) \leq \left( r - \frac{r}{16r^3+1} \right) n + o(n) \text{ as }n\rightarrow \infty.$$ The main tool is a stability version of the Erdős-Gallai theorem that may be of… 
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  • T. Luczak
  • Mathematics
    J. Comb. Theory, Ser. B
  • 1999
It is shown that the value of R(Cn, Cn , Cn ; Cn) does not grow with n much faster than 4n, and this problem is not able to settle.