Schur number $S(8)$

The largest number $n$ such that the integers $1$ to $n$ can be colored with $8$ colors with no monochromatic solution to $x + y = z$.

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: $5372$ Lean proof
  schur-8-lower-5372, submitted by darkhipo, 2026-11-04
  [construction from Z. Chen, L. Zhu and F. Long, A new lower bound for the sixth Schur number: S(6) >= 537, arXiv:2610.31362.]
• Upper bound: None yet

Updates from the literature

• 2019-12-03 Lower bound: $5286$
  F. Rowley, A generalised linear Ramsey graph construction, Australasian Journal of Combinatorics 81 (2021) 245-256, arXiv:1912.01164.
  [via R. Ageron, P. Casteras, T. Pellerin, Y. Portella, A. Rimmel and J. Tomasik, New lower bounds for Schur and weak Schur numbers, arXiv:2112.03175.]
• 2026-07-16 Lower bound: $5362$
  Bengone, Brouk, Grinsztajn, Helbert, Lugherini, Rimmel and Tomasik, Shifted S-templates and improved lower bounds for Schur numbers, arXiv:2607.15034.
• 2026-10-22 Lower bound: $5372$
  Z. Chen, L. Zhu and F. Long, A new lower bound for the sixth Schur number: S(6) >= 537, arXiv:2610.31362.