Mathematics
Ramsey Numbers and the Threshold for Unavoidable Monochromatic Cliques
Quick fact
The Ramsey number R(3,3) equals 6, meaning that any red/blue coloring of the edges of a complete graph on 6 vertices must contain a monochromatic triangle, yet there exists a coloring on 5 vertices that avoids any monochromatic triangle.
Why this is interesting
Among any six people, you will always find either three mutual acquaintances or three mutual strangers. Why is that always true?