Lower bound for monochromatic triangles on 2 coloring of $K_n$

Question

I am trying to show that there is at least $\frac{n(n-1)(n-5)}{24}$ monochromatic triangles in any 2 coloring of the edges of $K_n$. I am trying to show this using Mantel's theorem but I can't seem to find a direction to go.

I have tried to determine how many non triangles there are on some fixed vertex but I don't see where to take the proof from there.

Answer 1

Hint: upper bound the number of triples $(u, v, w)$ of vertices such that the edges $(u, v)$ and $(v, w)$ have different colors (this will happen when $v$ is incident to roughly the same number of red and blue edges). Each non-triangle is associated to 2 such triples, and each triple belongs to exactly one non-triangle.