Goodman's Formula
A two-coloring of a complete graph K_n of n nodes which contains exactly the number of monochromatic forced triangles and no more (i.e., a minimum of R+B where R and B are the number of red and blue triangles) is called an extremal graph. Goodman (1959) showed that for an extremal graph,
Schwenk (1972) rewrote the equation in the form
where (n; k) is a binomial coefficient and |_x_| is the floor function.
See also
Blue-Empty Graph, Extremal Graph, Monochromatic Forced TriangleExplore with Wolfram|Alpha
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References
Goodman, A. W. "On Sets of Acquaintances and Strangers at Any Party." Amer. Math. Monthly 66, 778-783, 1959.Schwenk, A. J. "Acquaintance Party Problem." Amer. Math. Monthly 79, 1113-1117, 1972.Referenced on Wolfram|Alpha
Goodman's FormulaCite this as:
Weisstein, Eric W. "Goodman's Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GoodmansFormula.html