6. Consider the complete graph Kn for n ≥ 3. Color r of the vertices in K, red and the remaining n − r(= g) vertices green. For any two vertices v, w in Kn, color the edge {v, w} (1) red if v, w are both red, (2) green if v, w are both green, or (3) blue if v, w have different colors. Assume that r≥ g. 1 a) Show that for r = 6 and g = 3 (and n = 9) the total number of red and green edges in K9 equals the number of blue edges in K9. b) Show that the total number of red and green edges in K, equals the number of blue edges in K, if and only if n = r+g, where 9, r are consecutive triangular numbers. [The triangular numbers are defined recursively by t₁ = 1,tn+1 tn = n(n+1)/2. Hence t₁ = 1, t2 = 3, t3 = 6,...] = = t + (n+1), n > 1; so

6. Consider the complete graph Kn for n ≥ 3. Color r of the vertices in Kn red and the
remaining n − r(= g) vertices green. For any two vertices v, w in Kn, color the edge {v, w} (1)
red if v, w are both red, (2) green if v, w are both green, or (3) blue if v, w have different colors.
Assume that r ≥ g.

a) Show that for r = 6 and g = 3 (and n = 9) the total number of red and green edges in K9
equals the number of blue edges in K9.
b) Show that the total number of red and green edges in Kn equals the number of blue edges
in Kn if and only if n = r + g, where g, r are consecutive triangular numbers.
[The triangular numbers are defined recursively by t1 = 1, tn+1 = tn + (n + 1), n ≥ 1; so
tn = n(n + 1)/2. Hence t1 = 1, t2 = 3, t3 = 6, . . . ]

PS: Please type with the correct one and do not use ChatGPT to give me the answer.