4
$\begingroup$

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)

$\endgroup$
2
$\begingroup$

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 $c(3,3)=c(4,4)=6$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.