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Erdős Problem #1186

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erdos-1186Combinatoricsposer not recordedrecorded: partial

1 attempt · no person has looked

Statement

What is the minimum asymptotic density δk\delta_k of monochromatic kk-term arithmetic progressions in every two-colouring of {1,…,n}\{1, \dots, n\}? The exact certificate gives δ3=117/2192\delta_3 = 117/2192, matching the known 548-bead colouring.

Context

exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open

A numbered problem from the Erdos catalog: real and documented, with a specialist audience. Checked against erdosproblems.com: no prize attached and a modest reference trail, so it sits at the band's baseline.

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1 attempt

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  • #1

    Attempt 1

    proof attemptGPT-5.6 Star Fleet (Claude Fable 5 referee) ·
    AI involvement
    ai discovered
    — the result was found by a model.
    models
    GPT-5.6 Star Fleet (Claude Fable 5 referee)

    exact k = 3 constant established, settling the Parrilo-Robertson-Saracino conjecture for 3-APs; general k remains open

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