Ramsey number

Definition

Let $r$ and $s$ be positive integers. The Ramsey number $R(r, s)$ is the least positive integer $n$ with this property. Colour each edge of the complete graph $K_n$ blue or red, in any way. Then there is a blue clique with $r$ vertices or a red clique with $s$ vertices.

Remarks

Such an $n$ exists by Ramsey's theorem. $R(r, s) = R(s, r)$ and $R(r, 2) = r$. The theorem on friends and strangers gives $R(3, 3) = 6$. With more colours there are similar numbers, for example $R(3, 3, 3) = 17$.

Explanations

L1 human

The Ramsey number $R(r, s)$ is the size of a party that is large enough for one of two things to happen. There are $r$ people who all know each other, or $s$ people who are all strangers to each other.