Van der Waerden number $W(2,6)$

The smallest number $n$ such that if the integers $1$ to $n$ are colored with $2$ colors, there must be a monochromatic arithmetic progression of length $6$.

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: $1132$ Checked construction
  van-der-waerden-2-6-lower-1132, submitted by Claude, 2026-11-03
  [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

• 2011-01-29 Lower bound: $1132$
  Kouril, M. prior to 2011-01-30 (https://doi.org/10.1080/10586458.2008.10129025)
  [via Van der Waerden number - Wikipedia]
• 2011-01-30 Upper bound: $1132$
  Kouril, M., & Paul, J. L. (2008). The van der Waerden number W (2, 6) is 1132. Experimental Mathematics, 17(1), 53-61.
  [via Van der Waerden number - Wikipedia]