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Erdős Problem #138

Erdős problem #138 · erdosproblems.com/138

If W(k)W(k) is the least NN such that every two-colouring of {1,…,N}\{1, \dots, N\} contains a monochromatic kk-term arithmetic progression, must W(k+1)−W(k)→∞W(k+1) - W(k) \to \infty?

Result
Proved(see note)
Status
Resolved
AI contribution
AI-discovered
Method
Argument
Field
Ramsey Theory
Posed by
—
Year posed
1981
Years open
45y
Solved
2026-05-21
Model
AlphaProof Nexus
Vendor
Google DeepMind
Collaborators
—
Verification
Lean-verified
Publication
Announced
Significance
14 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

the stronger question W(k)1/k→∞W(k)^{1/k} \to \infty remains open The stronger question, whether W(k)^(1/k) tends to infinity, is claimed in the OpenAI math release (September 2026); see erdos-138-superexponential-van-der-waerden.

What the AI did

Proved by AlphaProof Nexus with a Lean-checked argument.

Verification

Lean-checked; official Erdős problems record updated.

Sources

Changelog1 change
  • Rasmus Lindahlchanged What was actually shown from “the stronger question $W(k)^{1/k} \to \infty$ remains open” to “the stronger question $W(k)^{1/k} \to \infty$ remains open The stronger question, whether …”

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