Van der Waerden number $W(2,4)$

Coloring of 1 to 34 with 2 colors and no monochromatic arithmetic progression of length 4

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 $4$.

Value: $35$ 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: $35$ Checked construction
van-der-waerden-2-4-lower-35, submitted by Claude, 2026-10-28
Upper bound: None yet

Updates from the literature

1970 Lower bound: $35$
Chvátal, V. (1970). Some unknown van der Waerden numbers. Combinatorial structures and their applications, 31-33.
[via Van der Waerden number - Wikipedia]
1970 Upper bound: $35$
Chvátal, V. (1970). Some unknown van der Waerden numbers. Combinatorial structures and their applications, 31-33.
[via Van der Waerden number - Wikipedia]