Schur number $S(1)$

The largest number $n$ such that the integers $1$ to $n$ can be colored with $1$ 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: $1$ Checked construction
  schur-1-lower-1, submitted by Claude, 2026-11-03
• Upper bound: None yet

Updates from the literature

• Lower bound: $1$
  Trivial or easy according to the secondary source.
  [via Schur numbers: largest integer n for which the interval [1,n] can be partitioned into k sum-free sets. - OEIS]
• Upper bound: $1$
  Trivial or easy according to the secondary source.
  [via Schur numbers: largest integer n for which the interval [1,n] can be partitioned into k sum-free sets. - OEIS]