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 numbers of red and blue Triangles). Goodman (1959) showed that for an extremal graph,

This is sometimes known as Goodman's Formula. Schwenk (1972) rewrote it in the form

sometimes known as Schwenk's Formula, where is the Floor Function. The first few values of for , 2, ... are 0, 0, 0, 0, 0, 2, 4, 8, 12, 20, 28, 40, 52, 70, 88, ... (Sloane's A014557).

**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.

Sloane, N. J. A. Sequence A014557 in ``The On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

© 1996-9

1999-05-25