Van der Waerden number $W(6,3)$
The smallest number $n$ such that if the integers $1$ to $n$ are colored with $6$ colors, there must be a monochromatic arithmetic progression of length $3$.
Lower bound:
$226$
Upper bound:
Unknown
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: None yet
- Upper bound: None yet
Updates from the literature
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2018-03
Lower bound: $226$
Blankenship, T., Cummings, J., & Taranchuk, V. (2018). A new lower bound for van der Waerden numbers. European Journal of Combinatorics, 69, 163-168.
[via Monroe, D. (2019). New Lower Bounds for van der Waerden Numbers Using Distributed Computing. arXiv preprint arXiv:1603.03301v6.]