Weak Schur number $WS(5)$

The largest number $n$ such that the integers $1$ to $n$ can be colored with $5$ 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: $207$ Checked construction
  weak-schur-5-lower-207, submitted by Claude, 2026-11-03
  [construction from F. Rowley, comment with a partition of [1,207] into 5 weakly sum-free sets, Sequence A072842 in the OEIS, 2025-08-29.]
• Upper bound: None yet

Updates from the literature

• 2006 Lower bound: $189$
  P.F. Blanchard, F. Harary and R. Reis, Partitions into sum-free sets, Integers 6 (2006) #A7.
  [via Weak Schur numbers: largest integer n for which the interval [1,n] can be partitioned into k weakly sum-free sets. - OEIS]
• 2011-11 Lower bound: $196$
  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 Weak Schur numbers: largest integer n for which the interval [1,n] can be partitioned into k weakly sum-free sets. - OEIS]
• 2025-08-29 Lower bound: $207$
  F. Rowley, comment with a partition of [1,207] into 5 weakly sum-free sets, Sequence A072842 in the OEIS, 2025-08-29.
  [via Weak Schur numbers: largest integer n for which the interval [1,n] can be partitioned into k weakly sum-free sets. - OEIS]