Erdős Problem #1186

Erdős problem #1186 · erdosproblems.com/1186

What is the minimum asymptotic density $\delta_k$ of monochromatic $k$-term arithmetic progressions in every two-colouring of $\{1, \dots, n\}$? The exact certificate gives $\delta_3 = 117/2192$, matching the known 548-bead colouring.

Result

Proved(see note)

Status

Partial result

AI contribution

AI-discovered

Method

Argument

Field

Arithmetic Ramsey Theory

Posed by

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Year posed

—

Years open

—

Solved

2026-07-13

Model

GPT-5.6 Star Fleet (Claude Fable 5 referee)

Vendor

OpenAI / Anthropic

Collaborators

—

Verification

Unreviewed

Publication

Announced

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article