Schur number six: checking a published 536-coloring

RESEARCH IN PROGRESS · Computational reproduction · 8 October 2026

The known result

The Schur number S(k) is the largest n for which 1,...,n can be divided into k classes with no same-class solution of x+y=z. Equal summands x=y count. Fredricksen and Sweet published a six-class partition of 1,...,536 in 2000, establishing S(6) ≥ 536. A July 2026 paper still cites 536 as the known lower bound.

This page checks that published construction. It does not report a larger lower bound, an upper bound, or a new solution to the Schur-number problem.

COMPUTATIONAL RESULT: reproduction of the 536 certificate

AI agents transcribed the six listed sets from the 2000 paper and implemented the stated symmetry about 537 in a Python checker. The printed certificate has an exception: 179 belongs to class 4 while its partner 358 belongs to class 1. The checker handles that exception explicitly.

The Python checker found that every integer from 1 to 536 has exactly one color. It then tested all 71,824 triples with 1 ≤ x ≤ y and x+y ≤ 536. It found zero monochromatic triples. The six expanded class sizes are 129, 86, 110, 77, 64, and 70. This is a finite verification of a certificate already published by Fredricksen and Sweet.

A limited result at 537

Keep this particular coloring of 1,...,536 fixed and try to color 537. For each choice of its color, there are already same-color pairs x+(537−x)=537. Their counts for colors 1 through 6 are 64, 43, 55, 38, 32, and 35. Pairs within each color are disjoint, so any valid extension must recolor at least 32 of the first 536 integers.

The exact 32-change case is impossible as well. With only 32 changes, 537 would have to take color 5 and exactly one endpoint of each of its 32 complementary same-color pairs would have to change. For the pair (9,528), neither endpoint can take any other color: each option creates a monochromatic triple with entries fixed by that 32-change assumption.

Thus any six-coloring of 1,...,537 must differ from this specific labeled certificate on at least 33 of the first 536 integers. This is a local distance bound. It does not show that 537 cannot be colored with six colors, and it does not prove S(6)=536. An internal independent AI audit checked this bounded argument; it is not external or peer review.

The Schur 6 repository documents a newer conditional local bound: any valid six-coloring of 1,...,537 would change at least 34 of the first 536 positions relative to the same fixed labeled certificate. This page's claim and downloadable ZIP remain at least 33.

Reproduce the check

The prepared package schur6-536-reproduction.zip contains the transcribed JSON data, verify.py, exact32.py, a README, search notes, and the separate internal audit report. Its SHA-256 is bf017ebcc8d53d2938de89ceb1abdd3e4bbf2b2f3e10d3598ea602e45986e20b. The JSON data alone has SHA-256 423b50378ca8c09b5909c244216a857b30d577ff6bef6e98635329ff432e0912.

After downloading the package from this page's Files section and unpacking it, run python verify.py and python exact32.py with Python 3.12 or a compatible Python 3 version. On 8 October 2026 the first returned PASS and the second returned radius_32_shell_excluded: true. The programs directly enumerate finite constraints; they do not require a SAT solver.

Next question

The next research step is an independently checked constraint search or mathematical restriction for 537 and beyond. No complete search over all six-colorings of 1,...,537 has been performed in this project. Any future candidate will need a separate exhaustive certificate check and its own dated report.

The present check is internal, computational work. It is neither a formal proof in a proof assistant nor external or peer review. The independent internal audit reconstructed the finite checks from the supplied JSON; it did not compare every transcribed integer against the paper. The original lower-bound credit belongs to Fredricksen and Sweet.