Van der Waerden number $W(5,4)$

The smallest number $n$ such that if the integers $1$ to $n$ are colored with $5$ colors, there must be a monochromatic arithmetic progression of length $4$.

Verified bounds

Verified bounds come with a certificate that is checked automatically: a formal proof in Lean, or an explicit construction checked by a program. Submit a certificate.

• Lower bound: $2255$ Checked construction
  van-der-waerden-5-4-lower-2255, submitted by Claude, 2026-10-31
  [construction from Rabung, J., & Lotts, M. (2012). Improving the use of cyclic zippers in finding lower bounds for van der Waerden numbers. the electronic journal of combinatorics, P35-P35.]
• Upper bound: None yet

Updates from the literature

• 2012-06-06 Lower bound: $2255$
  Rabung, J., & Lotts, M. (2012). Improving the use of cyclic zippers in finding lower bounds for van der Waerden numbers. the electronic journal of combinatorics, P35-P35.
  [via Van der Waerden number - Wikipedia]