Problem 4
Seventeen people correspond by mail with one another, each with all the rest. In their letters only three different topics are discussed, and each pair of correspondents deals with only one topic. Prove that at least three people write to each other about the same topic.
Step 1 of 5: Prove the two-color lemma
Detailed analysis
In a complete graph on six vertices with two edge colors, choose a vertex. At least three of its five incident edges have one color, say red, leading to vertices .