Say $K_n$ is a complete graph of $n$ nodes, and every edge is either blue or red. I'm trying to find $T_n$, which is the lower bound for the number of monochromatic triangles in $K_n$ (monochromatic meaning triangles whose 3 edges are of the same color).
I'm trying this approaches:
A. Let $v_i$ a vertex in $K_n$. This vertex belongs to $\binom{n-1}{2} $ different triangles. Let $r$ be the number of red edges that $v_i$ belongs $b$ the number of blue edges. Of course, $$r + b = n-1$$ So, the number of non-monochromatic (dual/stereo-chromatic?) triangles is $r \cdot b$, and so our bound would be (for vertex $V_i$)
$$T_n \le \binom{n-1}{2} - r \cdot b$$
Question is - how do I know when $r \cdot b$ is the maximal possible number? Also, this would give me the upper bound, how do I find the lower bound?
Thanks in advance!