I do not know of any solution or if it's an open problem: Let $R(i,i)=k$, therefore there exists a counter examples with blue and red edges for a clique of size $k-1$. Does there exist a counter-example for $k-1$ nodes with an almost equal number of red and blue edges (any bound is welcome). Even more interesting: Is there a counter-example with a not almost equal number of red and blue edges for $k-1$ nodes? Define almost equal number however you want, I am interested in any result.
As a user suggested in the comments, I am also interested in Ramsey numbers on any graph, not just Ki
$K_9$ cannot be decomposed into two planar graphs.