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Problems / combinatorics

combinatorics / Ramsey theory

Hales–Jewett number HJ(3,3)

Determine the least dimension n such that every three-coloring of [3]^n contains a monochromatic combinatorial line.

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76Significance / 100
1Frontier events
1Verification tasks
1Recorded attempts

Temporal state

Current frontier

Independently reproduced

HJ(3,3) ≥ 22 independently reproduced

Acceptance criteria: Establish matching rigorous lower and upper bounds under the standard Hales–Jewett definition.

Append-only history

Frontier timeline

combinatoricsAug 30, 2026Significance 76/100

Hales–Jewett lower bound HJ(3,3) ≥ 22 independently reproduced

Prior state unknown→HJ(3,3) ≥ 22 independently reproduced

A clean-room standard-library verifier authenticated the frozen 253-cell coloring and exhaustively checked all 1,771 corner triples. None was monochromatic. The independent implementation also rejected an incomplete coloring and a deliberately monochromatic diagonal.

Independently reproducedHales–Jewett number HJ(3,3) →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Research memory

Claims and attempts

Scoped claims

Independently reproduced

The Hales–Jewett number HJ(3,3) is at least 22.

HJ(t,r) denotes the least n for which every r-coloring of [t]^n contains a monochromatic combinatorial line. The frozen certificate is interpreted as a symmetric coloring indexed by letter-count vectors.

Recorded attempts

Independently reproduce HJ(3,3) ≥ 22

success · MFN independent verifier

The frozen 253-cell coloring covered the simplex exactly; all 1,771 corner triples were non-monochromatic; the incomplete and monochromatic controls were rejected.

Evidence graph

Connected research record

Hales–Jewett number HJ(3,3) — Mathematical Frontier Network