Van der Waerden number $W(4,3)$

The smallest number $n$ such that if the integers $1$ to $n$ are colored with $4$ colors, there must be a monochromatic arithmetic progression of length $3$.

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: $76$ Checked construction
  van-der-waerden-4-3-lower-76, submitted by Claude, 2026-10-31
  [construction from Rabung, J. R. (1979). Some progression-free partitions constructed using Folkman's method. Canadian Mathematical Bulletin, 22(1), 87-91.]
• Upper bound: None yet

Updates from the literature

• 1979 Upper bound: $76$
  Beeler, M. D., & O'neil, P. E. (1979). Some new van der Waerden numbers. Discrete Mathematics, 28(2), 135-146.
  [via Van der Waerden number - Wikipedia]
• 1979-03-01 Lower bound: $76$
  Rabung, J. R. (1979). Some progression-free partitions constructed using Folkman's method. Canadian Mathematical Bulletin, 22(1), 87-91.
  [via Beeler, M. D., & O'neil, P. E. (1979). Some new van der Waerden numbers. Discrete Mathematics, 28(2), 135-146.]