Problem
COM-B2-M06-P007 Six People
#7
★★★☆☆ Level 3 of 5
Prove that among any \(6\) people there are either \(3\) pairwise acquainted people or \(3\) pairwise unacquainted people. Assume acquaintance is mutual.
Acquaintance and non-acquaintance are two colours of the edges of a complete graph.
Build a graph on \(6\) vertices: the vertices are people. Colour an edge red if the two people are acquainted, and blue if they are not.
By the \(K_6\) result, this colouring contains a monochromatic triangle. A red triangle means \(3\) pairwise acquainted people, and a blue triangle means \(3\) pairwise unacquainted people.
Students should get used to the translation “people → graph”.