TITLE: Reference request: monochromatic paths in edge-colored complete graphs
QUESTION [7 upvotes]: Given $k,c \in \mathbb{N}$, let $P(k,c)$ be the minimum $n$ such that no matter how we color the edges of the complete graph $K_n$ with $c$ colors, there is always a monochromatic path of length $k$.
What are the best known upper and lower bounds for $P(k,c)$?

REPLY [4 votes]: For the multi-color Ramsey numbers of even cycles, Luczak, Simonovits and Skokan proved that $R(C_k;c)\le ck+o(k)$ for fixed number $c$ of colors and $k\rightarrow \infty$.
For odd cycles, Bondy and Erdos claim that $R(C_k;c)\le (c+2)!k$.
Both upper bounds apply also for paths with $k$ vertices since $P(k-1,c) \le R(C_k;c)$.