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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


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

© 1996-9 Eric W. Weisstein