Van der Waerden number $W(2,3)$

Coloring of 1 to 8 with 2 colors and no monochromatic arithmetic progression of length 3

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 $3$.

Value: $9$ 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: $9$ Checked construction
van-der-waerden-2-3-lower-9, submitted by Claude, 2026-10-28
Upper bound: None yet

Updates from the literature

1970 Lower bound: $9$
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: $9$
Chvátal, V. (1970). Some unknown van der Waerden numbers. Combinatorial structures and their applications, 31-33.
[via Van der Waerden number - Wikipedia]