It is well known that the minimum number of monochromatic triangles in a red/blue coloring of the edges of the complete graph $K_n$ is given by Goodman's formula $$M(n)=\binom n3-\left\lfloor\frac n2\left\lfloor\left(\frac{n-1}2\right)^2\right\rfloor\right\rfloor;$$ see OEIS sequence A014557 or the original paper by A. W. Goodman, On sets of acquaintances and strangers at any party, Amer. Math. Monthly 66 (1959), 778–783, or my answer to this math.stackexchange question.

Is there any literature on the more general question, what is the minimum number of monochromatic triangles in a red/blue coloring of the edges of the complete graph $K_n$

with a prescribed number of edges of each color?

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Is there any literature on the related question, given natural numbers $n$ and $k$, what is the minimum value of the quantity $m+kb$ over all red/blue colorings of the edges of the complete graph $K_n$, where $m$ is the number of monochromatic triangles (of either color) and $b$ is the number of blue edges?