A two-coloring of a complete graph of nodes which contains exactly the number of monochromatic
forced triangles and no more (i.e., a minimum of where and are the number of red and blue triangles)
is called an extremal graph. Goodman (1959) showed
that for an extremal graph,

Goodman, A. W. "On Sets of Acquaintances and Strangers at Any Party." Amer. Math. Monthly66, 778-783, 1959.Schwenk,
A. J. "Acquaintance Party Problem." Amer. Math. Monthly79,
1113-1117, 1972.