Hello,


I am wondering, if we have a complete graph on $n$ vertices, and we have $k$ colours so that every edge of the graph is coloured with one of these colours, what is the least $n$ such that we will always be able to find a monochromatic cycle of length $m$?


It would be great to find a function F($k,m$) to give such a least $n$, or at least find good lower and upper bounds on it.



Looking for monochromatic cycles in an edge-coloured clique
You in the realms of extensions of Ramsey Theory here, so general, precise answers may be a bit thin on the ground.


There are at least a couple of known results:


$2^{k} \leq F(k,3) \leq 3k!$


and


$F(k,4) \leq k^{2} + k + 1$


These lecture slides give a start:
<https://www.dpmms.cam.ac.uk/~dc340/Ramsey-course.html>


