# Ramsey number R(5,5)

- kind: problem
- status: open
- year: 1955

**Statement.** What is the smallest $n$ such that every red/blue coloring of the edges of $K_n$ has a monochromatic $K_5$?

It is known that $43 \le R(5,5) \le 46$, with the upper bound brought down in 2024 by Angeltveit and McKay. Erdős joked that if aliens demanded R(5,5) we should marshal all our computers, and if they demanded R(6,6) we should try to destroy them. The asymptotic growth rate improved in 2023, when Campos, Griffiths, Morris and Sahasrabudhe beat the $4^n$ barrier.

## What would settle it

Lean statement in formal-conjectures; any exact value is a finite check

## Sources

- [Wikipedia: Ramsey's theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem)

[Combinatorics](/wiki/Space:combinatorics.md) · [Mathematics](/wiki/Continent:mathematics.md)

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