Schur number $S(6)$

The largest number $n$ such that the integers $1$ to $n$ can be colored with $6$ 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: $537$ Checked construction
  schur-6-lower-537, submitted by Claude, 2026-11-03
  [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

• 2000-05-18 Lower bound: $536$
  H. Fredricksen and M.M. Sweet, Symmetric Sum-Free Partitions and Lower Bounds for Schur Numbers, Electronic Journal of Combinatorics 7 (2000) #R32.
  [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.]
• 2026-10-22 Lower bound: $537$
  Z. Chen, L. Zhu and F. Long, A new lower bound for the sixth Schur number: S(6) >= 537, arXiv:2610.31362.