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

Coloring of 1 to 292 with 3 colors and no monochromatic arithmetic progression of length 4

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

Value: $293$ Verified lower bound

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: $293$ Checked construction
van-der-waerden-3-4-lower-293, 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

2012 Lower bound: $293$
Kouril, Michal (2012). Computing the van der Waerden number W(3,4)=293. Integers. 12: A46.
[via Van der Waerden number - Wikipedia]
2012 Upper bound: $293$
Kouril, Michal (2012). Computing the van der Waerden number W(3,4)=293. Integers. 12: A46.
[via Van der Waerden number - Wikipedia]