Problem
COM-B2-M06-P012 Seven People and an Extra Vertex
#12
★★★☆☆ Level 3 of 5
Prove that among any \(7\) people there are either \(3\) pairwise acquainted people or \(3\) pairwise unacquainted people. Explain why \(7\) is not the exact threshold.
Choose any \(6\) of the \(7\) people.
Among any chosen \(6\) of the \(7\) people, by \(R(3,3)=6\), there are already either \(3\) pairwise acquainted people or \(3\) pairwise unacquainted people. Hence such a triple also exists among all \(7\) people.
The number \(7\) is not the exact threshold because the same result is already guaranteed for \(6\) people.
Useful for understanding the difference between a sufficient and a minimal number.