Schwenk's Formula

Let $R+B$ be the number of Monochromatic Forced Triangles (where $R$ and $B$ are the number of red and blue Triangles) in an Extremal Graph. Then

$$R+B={n\choose 3}-\left\lfloor{{\textstyle{1\over 2}}n\left\l... ...r{{\textstyle{1\over 4}}(n-1)^2}\right\rfloor }\right\rfloor ,$$

where ${n\choose k}$ is a Binomial Coefficient and $\left\lfloor{x}\right\rfloor$ is the Floor Function (Schwenk 1972).

See also Extremal Graph, Monochromatic Forced Triangle

References

Schwenk, A. J. ``Acquaintance Party Problem.'' Amer. Math. Monthly 79, 1113-1117, 1972.