Invite six people to a party and three of them are guaranteed to be mutual friends or mutual strangers — no matter who you invite. That is the smallest case of a theorem saying complete disorder is impossible: any structure large enough must contain the pattern you were trying to avoid. Across 40 interactive demonstrations, try and fail to two-colour six vertices, split sixteen vertices three ways with an exceptional graph, and follow bounds that did not move between 1935 and 2023.
See also: Graph Theory for colouring and cliques, Probability for the method Erdős invented here, and Sphere Packing for another problem where a human and a machine moved decades-old bounds a year apart.
Six people, and three of them are always mutual friends or mutual strangers
Order is unavoidable — every large enough structure contains the pattern you were avoiding
Erdős proved colourings exist without ever building one, and invented a whole technique doing it
We know R(4,4) is 18 and have almost no idea what R(5,5) is
Sixteen vertices, three colours, no monochromatic triangle — and one exceptional graph behind it
Colour the whole number line and arithmetic patterns survive anyway
Scatter enough points and a convex polygon appears whether you want one or not
The 2023 result that moved a bound nobody had improved since 1935
With k colours the answer explodes — and for decades nobody could say how fast
A $100 question of Erdős, answered in 2026 — and the answer is no