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The Abbott-Hanson Recurrence for Schur Numbers

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schur-number-recurrenceCombinatoricsposed by Harvey Abbott, Denis Hanson, 1972recorded: partial

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Statement

The classical Abbott-Hanson recurrence gives S(k+2)≥9S(k)+4S(k+2) \ge 9S(k)+4 for Schur numbers, and had stood as the basis for the best asymptotic lower bounds. Shifted SS-templates, a more flexible form of Rowley's template construction, yield S(k+2)≥10S(k)+2S(k+2) \ge 10S(k)+2 and hence improved lower bounds.

Context

improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown

Schur numbers are a classical Ramsey-theoretic quantity where progress is rare and the Abbott-Hanson recurrence had been the standard tool for over fifty years.

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  • #1

    Attempt 1

    constructionChatGPT 5.5 Pro with Nils Bengone, Amine Brouk, Max Grinsztajn, Terence Helbert, Bao Lugherini, Arpad Rimmel, Joanna Tomasik ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    ChatGPT 5.5 Pro
    people
    Nils Bengone, Amine Brouk, Max Grinsztajn, Terence Helbert, Bao Lugherini, Arpad Rimmel, Joanna Tomasik

    The abstract says the shifted SS-template construction was discovered during a conversation with ChatGPT 5.5 Pro and then refined, verified and extended to multiple templates by the authors, and the conclusion describes the paper as building on an original idea of the model.

    improves the classical recurrence; the exact Schur numbers beyond S(5) remain unknown

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