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New Bounds for Double Covers of the Discrete Box

Combinatorics · posed by Leader, Miličević and Tan · partial

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Statement

For A={0,1,2}dA=\{0,1,2\}^d, write f(d)f(d) for the fewest proper sub-boxes covering every point exactly twice. Leader, Miličević and Tan asked whether f(d)≥2df(d)\ge 2^d for all dd, as Question 4.1 of the PatternBoost paper. The paper gives new bounds on f(d)f(d).

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    Patrick White, using Claude

    That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.

    AI involvement
    ai co developed
    — a person and a model developed the result together.

    The author credits large language models used as reasoning engines with producing several of the key ideas, naming the modular refinement of Section 3, the crossing lemma of Section 5 and the constructions of Section 7.

    Improved bounds rather than a settled question: the asked-for inequality is not established in general.

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