Weak Schur number $WS(6)$

The largest number $n$ such that the integers $1$ to $n$ can be colored with $6$ colors with no monochromatic solution to $x + y = z$ with $x \neq y$.

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: $650$ Checked construction
  weak-schur-6-lower-650, submitted by Claude, 2026-11-03
  [construction from S.E. Rodrigo, A new lower bound for the sixth weak Schur number: WS(6) >= 650, Zenodo preprint.]
• Upper bound: None yet

Updates from the literature

• 2011-11 Lower bound: $572$
  S. Eliahou, J.M. Marín, M.P. Revuelta and M.I. Sanz, Weak Schur numbers and the search for G. W. Walker's lost partitions, Computers & Mathematics with Applications 63 (2012) 175-182.
  [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.]
• 2012 Lower bound: $574$
  D. Robilliard, C. Fonlupt, V. Marion-Poty and A. Boumaza, A Multilevel Tabu Search with Backtracking for Exploring Weak Schur Numbers, Artificial Evolution (EA 2011), Lecture Notes in Computer Science 7401 (2012) 109-119.
  [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.]
• 2012-07 Lower bound: $581$
  R. Le Bras, C.P. Gomes and B. Selman, From Streamlined Combinatorial Search to Efficient Constructive Procedures, Proceedings of the AAAI Conference on Artificial Intelligence 26 (2012) 499-506.
  [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.]
• 2013-04 Lower bound: $582$
  S. Eliahou, C. Fonlupt, J. Fromentin, V. Marion-Poty, D. Robilliard and F. Teytaud, Investigating Monte-Carlo Methods on the Weak Schur Problem, EvoCOP 2013, Lecture Notes in Computer Science 7832 (2013) 191-201.
  [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.]
• 2020-11-23 Lower bound: $642$
  F. Rowley, New lower bounds for weak Schur partitions, Integers 21 (2021) #A59, arXiv:2011.11292.
  [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.]
• 2021-12-06 Lower bound: $646$
  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.
• 2026-06-10 Lower bound: $650$
  S.E. Rodrigo, A new lower bound for the sixth weak Schur number: WS(6) >= 650, Zenodo preprint.