In general, an extremal graph is the largest graph of order which does not contain a given graph as a subgraph (Skiena 1990, p. 143).
Turán studied extremal graphs that do not contain a complete
graph
as a subgraph.

One much-studied type of extremal graph is 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 of this type,

(1)

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

(2)

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, ... (OEIS A014557).

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.Skiena, S. Implementing
Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading,
MA: Addison-Wesley, p. 143, 1990.Sloane, N. J. A. Sequence
A014557 in "The On-Line Encyclopedia
of Integer Sequences."