Van der Waerden number $W(5,6)$
The smallest number $n$ such that if the integers $1$ to $n$ are colored with $5$ colors, there must be a monochromatic arithmetic progression of length $6$.
Lower bound:
$786741$
(786,741)
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
-
Lower bound: $540026$
(540,026)
Reference unknown
[via Blankenship, T., Cummings, J., & Taranchuk, V. (2018). A new lower bound for van der Waerden numbers. European Journal of Combinatorics, 69, 163-168.] -
2019-05-22
Lower bound: $786741$
(786,741)
Monroe, D. (2019). New Lower Bounds for van der Waerden Numbers Using Distributed Computing. arXiv preprint arXiv:1603.03301v6.
[via Van der Waerden number - Wikipedia]