Ramsey's theorem

The problem is classic, given 6 people, either there is a group of 3 who know each other, or there is a group of 3 who do not know each other, assuming the known relationship is bidirectional.

Proof from West's Graph Theory book.
Use pigeon-hole principle, among 6 nodes, in G and ~G, at least one of them have deg 3, because deg_G(v) + deg_~G(v) = 5. Let's say it is G, and node v has deg >= 3. Now we consider 3 neighbors of v, none of them have an edge between them, otherwise we have a triangle. It follows that the three neighbors form a triple with no edges.