Least monochromatic triangles in any two-colouring of K_n

Maximising \(\sum_v r_v b_v\) subject to \(r_v + b_v = n-1\) in Goodman's identity gives $$\#mono \ge \binom{n}{3} - \frac{n}{2} \left\lfloor \frac{(n-1)^2}{4} \right\rfloor .$$

Usage

morie_goodman_minimum(n)

Arguments

n

Number of vertices, at least 3.

Value

A list with minimum, total_triangles, max_bichromatic.

References

Goodman AW (1959) Amer Math Monthly 66(9):778-783.

Examples

morie_goodman_minimum(n = 5L)
#> $minimum
#> [1] 0
#> 
#> $total_triangles
#> [1] 10
#> 
#> $max_bichromatic
#> [1] 10
#>