Working on a take home test and have been stuck on this one proof for hours. Just can't seem to find a good logic behind it. Here it is.

I have to prove that any graph G=(V,E) with |V|=6 is either 3-complete or 3-isolated. If you are unsure of what 3-complete and 3-isolated are, here are the definitions:

A graph G=(V,E) is called k-complete if it has a subset, V', of its vertices with the number of vertices = k such that for any pair of vertices v not equal to w in V', v and w are adjacent.

A graph G=(V,E) is called k-isolated if it has a subset, V', of its vertices with the number of vertices = k such that for any pair of vertices v not equal to w in V', v and w are not adjacent.

Thanks in advance!