Problem
COM-B1-M05-P022 Five Around One Person
#22
★★★★☆ Level 4 of 5
In a group of \(10\) people, prove that there is a person who has either \(5\) acquaintances or \(5\) strangers.
Take any person and look at the other \(9\).
Choose any person \(A\). Among the other \(9\), each is either acquainted with \(A\) or not. There are two boxes: acquaintances and strangers. By the strengthened pigeonhole principle, one contains at least \(\lceil9/2 ceil=5\) people. Thus \(A\) has \(5\) acquaintances or \(5\) strangers.
Preparation for Ramsey reasoning.