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Problems / combinatorics

combinatorics / Ramsey theory

Hales–Jewett number HJ(4,2)

Determine the least dimension n such that every two-coloring of [4]^n contains a monochromatic combinatorial line.

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72Significance / 100
1Frontier events
1Verification tasks
1Recorded attempts

Temporal state

Current frontier

Independently reproduced

HJ(4,2) ≥ 14 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 72/100

Hales–Jewett lower bound HJ(4,2) ≥ 14 independently reproduced

Prior state unknown→HJ(4,2) ≥ 14 independently reproduced

A clean-room standard-library verifier authenticated the frozen 560-cell coloring and exhaustively checked all 1,820 corner quadruples. None was monochromatic. The independent implementation also rejected an incomplete coloring and a deliberately monochromatic diagonal.

Independently reproducedHales–Jewett number HJ(4,2) →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Research memory

Claims and attempts

Scoped claims

Independently reproduced

The Hales–Jewett number HJ(4,2) is at least 14.

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(4,2) ≥ 14

success · MFN independent verifier

The frozen 560-cell coloring covered the simplex exactly; all 1,820 corner quadruples were non-monochromatic; the incomplete and monochromatic controls were rejected.

Evidence graph

Connected research record