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.



Combinatorial optimization and graph coloring
Linear lower bound. For any 18 vertices we have a monochromatic quadrilateral by Ramsey theorem. It follows that the total number of monochromatic quadrilaterals is not less then ${n\choose 18}/{n-4\choose 14}$. Then one of triangles is contained in at least at least $4{n\choose 18}/({n-4\choose 14}\cdot {n\choose 3})=(n-3)/{18\choose 4}$ monochromatic quadrilaterals. 


