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 3 of 5: Choose six same-colored neighbors
Detailed analysis
Color the edge joining each pair of people by their topic. From any person , the sixteen incident edges use three colors, so at least six go to vertices by the same color, say color 1.