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).
