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Improved Bound for Colorings Without Symmetrically Colored k-APs

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symmetrically-colored-k-ap-boundCombinatoricsposed by Deng, Tidor and Zhaorecorded: partial

1 attempt · no person has looked

Statement

Deng, Tidor and Zhao asked whether [N][N] admits a coloring with No(1)N^{o(1)} colors and no symmetrically coloured 4-term arithmetic progression, giving an O(Nlog⁡223)O(N^{\log_{22}3}) coloring. The paper gives an Ok(N4/k2)O_k(N^{4/k^2}) coloring of [N][N] avoiding symmetrically coloured kk-APs for every even k≥4k\ge4, improving the exponent.

Context

Improves the exponent rather than answering the asked question: whether an No(1)N^{o(1)} colouring exists remains open.

Progress on a question of Deng, Tidor and Zhao, improving the known exponent for every even k at once.

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1 attempt

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  • #1

    Attempt 1

    constructionGPT-5 with Ruizhe Shi, Yiqi Dong ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5
    people
    Ruizhe Shi, Yiqi Dong

    The acknowledgements state the key construction idea was suggested during an interaction with GPT-5.

    Improves the exponent rather than answering the asked question: whether an No(1)N^{o(1)} colouring exists remains open.

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