I am considering the following problem:

(i) Fix $n$ and color the edges of $K_n$ red and blue arbitrarily.

(ii) Let $M$ be the set of monochromatic triangles in $K_n$ and define $g:M\rightarrow \mathbb{N}$ as $g(T_{xyz})$= $|N_r(x)\cap N_r(y)\cap N_r(z)|$ if $T_{xyz}$ is red and $|N_b(x)\cap N_b(y)\cap N_b(z)|$ otherwise. Here $T_{xyz}$ is a monochromatic triangle with vertices $x$, $y$ and $z$.

(iii) Consider the problem Max $g(T_{xyz})$ s.t. $T_{xyz}\in M$. What is a lower bound for this maximum?

Any advice as to how this problem may be tackled would be much appreciated.

Thanks

Update: As per a conjecture of A. Thomason (1979) $r(K_m+\bar{K_n})\le 2^m(m+n-2)+2$ where $r(G)$ is the diagonal ramsey number.