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The Order of Long Rainbow Arithmetic Progressions

Combinatorics · posed by Veselin Jungic, Jacob Licht, Mohammad Mahdian, Jaroslav Nesetril, Rados Radoicic, 2003 · disproved

1 attempt

Statement

Let TkT_k be the least tt such that every equinumerous tt-coloring of [tn][tn] contains a rainbow kk-term arithmetic progression. Jungic, Licht, Mahdian, Nesetril and Radoicic conjectured Tk=Θ(k2)T_k = \Theta(k^2); Conlon, Fox and Sudakov proved Tk=O(k2log⁡k)T_k = O(k^2 \log k). The matching lower bound Tk=Ω(k2log⁡k)T_k = \Omega(k^2 \log k) holds, so Tk=Θ(k2log⁡k)T_k = \Theta(k^2 \log k) and the conjectured order is wrong.

Context

the true order is determined, and it is not the conjectured one

A 2003 Combinatorics, Probability and Computing conjecture in anti-Ramsey theory that Conlon, Fox and Sudakov had already worked on, well known within extremal combinatorics.

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  • #1

    Attempt 1

    constructionCodex (GPT-5.6), Claude Code (Fable 5) with Jesse Geneson ·
    AI involvement
    ai assisted
    — a person led the work and used a model along the way.
    models
    Codex (GPT-5.6), Claude Code (Fable 5)
    people
    Jesse Geneson

    The acknowledgement says the two systems were used for proof exploration, proof criticism, exposition and revision. Proof exploration and criticism are mathematical work rather than prose work, but no specific step is attributed, so the lowest tier applies.

    the true order is determined, and it is not the conjectured one

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