Schur number six: the published 536 certificate and a local distance bound at 537

A local distance result · 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 check examined this bounded argument.

The Schur 6 repository contains the public source for the stronger local result proved below.

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.

Version 5 update: excluding 33 recolorings

The earlier argument established a local lower bound of 33 changes relative to this particular labeled certificate. The new question is whether exactly 33 changes could suffice. The same complementary-pair counts force color 5 at 537: every other color needs at least 35 changes. The 32 disjoint color-5 pairs require one changed endpoint each. Therefore at most one additional position outside their 64 endpoints can change. The extra change could instead alter a second endpoint, leaving every outside position fixed.

The forced pair (9,528) rules out both possibilities. Whichever endpoint changes, and whichever of the five other colors it receives, two different monochromatic triples appear. In each row below, all entries other than the changed endpoint lie outside the 64 forced-pair endpoints; the two sets of such entries are disjoint.

Repairing both blockers for any choice requires at least two changes outside the 64 endpoints, yet at most one is available. Thus a valid six-coloring of 1,...,537 must differ from this specific labeled certificate for 1,...,536 on at least 34 positions among 1,...,536. This does not show that an extension exists at distance 34, that any six-coloring of 1,...,537 exists, or that S(6)=536. It does not improve the known lower bound S(6) ≥ 536.

The new package schur6-536-radius33-reproduction.zip contains the earlier certificate and checks plus exact33.py, audit33.py, and audit-exact33.md. Its SHA-256 is 31bc175813971f6985a3b74f306bb51006e1885d84b18e1ec1739533ea57eae9. Download it from this page's Files section, unpack it, and run python exact33.py and python audit33.py. The former returns radius_33_shell_excluded: true; the latter returns PASS after checking all 20 explicit blocker triples. A separate internal review reconstructed the argument and checked all 20 listed witnesses. A subsequent source comparison checked all 269 printed certificate entries against the publisher’s PDF.

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.

This is internal computational work, without external peer review or formal proof-assistant verification. The original lower-bound result is due to Fredricksen and Sweet.