Lower and upper bounds for Schur numbers
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The Schur number S(k) is the largest N for which {1,…,N} admits a k-coloring with no monochromatic solution to x+y=z. How can recursive constructions and obstructions improve its lower and upper bounds?
Why it matters and what counts as progress
Why it matters: A better recurrence produces stronger finite bounds, while conditional upper bounds expose how Ramsey information constrains Schur colorings.
What counts as progress: A verified larger sum-free coloring, a stronger recursion, or a proved upper bound; any separate Ramsey premise must remain explicit.
How the problem developed
- 1972 · Lower bound [1]
H. L. Abbott and D. Hanson Improved recursive lower bounds for Schur numbers using generalized constructions.
- 26 Apr 2018 · Exact value [2]
Marijn J. H. Heule Determined the exact value S(5)=160 with a machine-checkable SAT proof.
- 16 Jul 2026 · Lower bound [3]
Luca Bengone, Julien Brouk, Pierre Grinsztajn, Luca Helbert, Alessio Lugherini, Pierre Rimmel and Martin Tomasik Improved the recursive lower bound to S(k+2)≥10S(k)+2, yielding S(8)≥5362 and S(13)≥2,038,282; the paper credits AI search with discovering the construction and human work with refinement.
AI contributions 2
AI agents derive a conditional bound of 1801 for the sixth Schur number
Conditional on R₄(3)≤61, the sixth Schur number satisfies S(6)≤1801; the general Ramsey-to-Schur bound and the stated frontier restrictions are proved with the Ramsey assertion left as an explicit premise.
Shifted S-templates and improved lower bounds for Schur numbers
An explicit width-10 shifted template gives S(k+2)≥10S(k)+2. Using existing bounds for S(6) and S(11), the authors obtain S(8)≥5362 and S(13)≥2038282.
Sources
Website timeline
Publication dates and editorial changes.
ChatGPT-origin shifted templates improve Schur number bounds
AI agents derive a conditional bound of 1801 for the sixth Schur number