A six-colouring of 29 points nobody had built
B(6) ≥ 29: the edges of K_29 partition into six graphs each with independence number ≤ 6 — a balanced 6-colouring in which every 7 vertices see all six colours. Equivalently R_6^5(K_7) ≥ 30.
First recorded movement on the r=6 case of Erdős problem 617 — the first r with no affine plane (Euler's 36 officers; Tarry 1900), so the classical construction behind every settled case (r=3,4,5,9, all prime powers) does not exist. Explicit witness colourings for K_26–K_29 are embedded in the claim script, which re-proves the theorem from scratch by enumerating every 7-subset (4,290,650 checks in total; 1,560,780 for K_29 alone). Construction: AG(2,5) direction-colouring plus up to four vertices whose stars satisfy a line-covering law — every colour-independent j-subset of new vertices must jointly contain j−1 full lines of that parallel class. Two independent routes (structural schema + SAT, and a joint-SAT climb) produced distinct witnesses at 28 and 29. Discovered 2026-08-09.
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Verification record — every quoted number, re-run 2026-08-09
| quantity | measured | verdict |
|---|---|---|
| B(6) lower bound | 29 | |
| subsets verified | 4,290,650 |
Provenance
script status: DETERMINISTIC
datasets:
re-run: 2026-08-09
Edges
Status, honestly
witness-verified by exhaustive enumeration (the strongest verification on this site: the claim script IS the proof). Novelty checked against nine sources 2026-08-09 — Chung–Liu 1978, Jacobson 1982, Erdős–Gyárfás 1999, Harborth–Möller 1999 (via nine citing works), Munemasa–Shinohara, Budden 2017, the June-2026 s-chromatic survey, and the erdosproblems.com 617 page/forum/proof-claims — none touch the six-colour diagonal. Residual: Budden's book ships 2026-08-14; HM99 PDF read directly only via citing works.. This status was assigned by hand from the public re-run audit above — it is a judgement made from that evidence, not a number this site computed for itself. What cannot move it is authority: no vote, endorsement, or say-so, from anyone including the maintainer. Independent reproduction would move it; nothing else will.
What this claim's own script draws. These are David Thompson's committed outputs — the same plots the script regenerates on any machine that runs it. You can redraw them yourself below.
Two runs of the same script — mine, and one you can start right now. A match proves the result is reproducible; it is still R0 on this registry's independence rings — same code, so it cannot move a status. Only an outside run does that.
USR-2026-0074 — B(6) >= 29: exhaustive witness verification
================================================================
K_26: 657,800 seven-subsets checked, 0 missing a colour -> OK [witness sha256:651b449a0ecf1903]
K_27: 888,030 seven-subsets checked, 0 missing a colour -> OK [witness sha256:20043d219b4deafc]
K_28: 1,184,040 seven-subsets checked, 0 missing a colour -> OK [witness sha256:fc799be8e2f4a31b]
K_29: 1,560,780 seven-subsets checked, 0 missing a colour -> OK [witness sha256:215b3891679a1c3d]
================================================================
ALL WITNESSES VERIFIED: the edges of K_29 partition into six graphs
of independence number <= 6. B(6) >= 29, i.e. R_6^5(K_7) >= 30.
Press Run it to execute this script in your own browser — real CPython, numpy and matplotlib compiled to WebAssembly. Nothing is sent to us; it runs on your CPU. First run downloads the runtime (~10 MB), then takes a few seconds. The script runs unmodified: every number printed here, and every figure it draws, is computed on your machine. The figures appear below.