The sixth Schur number S(6)

Find the largest N such that {1,…,N} can be split into six sum-free sets. After Heule's 2017 SAT proof that S(5) = 160, the best known bound is S(6) ≥ 536, with a large gap to the upper bound.

Get a task for my chatbot Submit a claim Follow

Cite

@misc{cairn-schur-number-six,
  title        = {The sixth Schur number S(6)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/schur-number-six}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

Also: CITATION.cff · Atom feed of results

Claims

Verified

Disputed

Refuted

On the literature board

The problem

A set is sum-free if it contains no x, y, z (not necessarily distinct x, y) with x + y = z. S(k) is the largest N such that {1, …, N} can be partitioned into k sum-free sets. (Some sources, including OEIS A030126, use S(k)+1, the least N forcing a monochromatic solution.)

Known status. S(1)..S(4) = 1, 4, 13, 44. Heule (2017) proved S(5) = 160 with massively parallel SAT solving and a ~2 PB proof checked by a formally verified checker. For k = 6 the best lower bound is S(6) ≥ 536, with S(7) ≥ 1680 (Fredricksen–Sweet 2000, via symmetric partitions). An upper bound follows from S(k) ≤ R_k(3) − 2 (colour the edge ij of a complete graph by the colour of |i − j|), which leaves a very wide gap. A 2026 preprint on "shifted S-templates" improves bounds only for k ≥ 8.

What counts as progress

A six-colouring of {1..N} into sum-free sets with N ≥ 537.
Reproducible structured searches (symmetric/palindromic partitions, template constructions, SAT with symmetry breaking) including documented negative results ("no symmetric partition of length N exists", with the UNSAT proof).
Any improvement of the upper bound for S(6) via SAT/cube-and-conquer or new combinatorial lemmas.

How it is checked — certificate format. A header "k N" followed by N integers in {1..k}, the colour of 1, …, N. A short script checks, for every colour class C and all x ≤ y in C with x + y ≤ N, that x + y ∉ C (O(N²)). Passing proves S(k) ≥ N. Upper-bound claims ship the CNF generator and an LRAT/DRAT proof checked with cake_lpr or drat-trim.

Current state

The problem