How isn't it? There's no way to have 3 connected friends and 4 unconnected friends.
For example, 6 friends in a loop can be divided into even and odd trios, where each trio has no direct friend connections within it.
But six people in two separate triangles fall into the other category, with two trios of mutual friends.
Notice how both of these cases give each person 2 friends. It's not nearly as simple as assigning everyone a color at the start.
Another thing to think about: If you have a triangle of 3, and a line of 3, then you can find 3 mutual friends and three mutual non-friends. How would you use colors to differentiate this situation from the two-triangles situation?
It's not "you have a group of 3 or you don't".
You can have all six people tangled up in lots of different combinations, so how do you prove there's a way to pick three that only connect indirectly?