1) what is the minimum $n$ such that in every $2$ - coloring of $K_n$ there exist a monochromatic copy of $C_m$ ?
2) moreover, what is the minimum $n$ such that in every $r$ - coloring of $K_n$ there exist a monochromatic cycle?
I found that the answer to 1 is known, but I don't find it. http://www.mate.unlp.edu.ar/~liliana/lawclique_2016/21.pdf - (page 24)
Let me elaborate on a comment of bof. For positive integers $r$ and $s$, let $c(r,s)$ denote the smallest positive integer $n$ such that if we color the edges of a clique on $n$ vertices with red and blue, either the red subgraph contains a cycle of length $r$ or the blue subgraph contains a cycle of length $s$. The main theorem of Faudree and Schelp [All Ramsey numbers for cycles in graphs, Discrete Math. 8 (1974), 313-329] is the following.
(i) If $3\le s\le r$ and $s$ is odd and $(r,s)\ne (3,3)$, then $c(r,s)=2r-1$.
(ii) If $4\le s\le r$ and $s$ and $r$ are even and $(r,s)\ne (4,4)$, then $c(r,s)=r+\frac12s+1$.
(iii) If $4\le s < r$ and $s$ is even and $r$ is odd, then $c(r,s)=\max(r+\frac12s-1,2s-1)$.
So the answer to (1), to guarantee a monochromatic copy of $C_m$, comes by setting $r=s=m$. To complete the answer, we mention that $c(3,3)=c(4,4)=6$.
