Fix $i,j$. I want a bound $F(i,j)$ on the following:

$\sum_{n=1}^\infty$ (the number of 2-colorings of the edges of $K_n$ such that the MAX RED clique is size exactly $i$ (that's an $i$ not a 1) and the MAX BLUE clique is of size exactly $j$.

Caveats:

1) If $n=3$ then coloring (1,2) RED, (2,3) RED, (1,3) BLUE is counted DIFF from coloring (2,3) RED, (3,1) RED, (1,2) BLUE.

2) The sum does not really go to infinity since when $n\ge$ RAMSEY$(i+1,j+1)$ it will be 0.

3) It might be that having MAX RED CLIQUE $\le i$ and MAX BLUE CLIQUE $\le j$ doesn't change things much and is easier to compute.