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Erdős Problem #856

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erdos-856Number theoryposed by Paul Erdős, 1970recorded: candidate

1 attempt · no person has looked

Statement

Let k≥3k\geq 3 and fk(N)f_k(N) be the maximum of ∑n∈A1n\sum_{n\in A}\frac{1}{n} over all A⊆{1,…,N}A\subseteq\{1,\ldots,N\} containing no kk subsets with the same pairwise least common multiple. Estimate fk(N)f_k(N). The claimed answer: fk(N)=(log⁡N)γk+o(1)f_k(N)=(\log N)^{\gamma_k+o(1)}, where γk\gamma_k is a weighted generalization of the Tang-Zhang sunflower capacity.

Context

Identifies the exponent as a variational sunflower-capacity constant, sharpening the Tang-Zhang bounds; the value of that constant itself remains open, as does site acceptance

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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Attempts

1 attempt

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  • #1

    Attempt 1

    proof attemptGPT-5.4 Pro with Przemysław Chojecki ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.4 Pro
    people
    Przemysław Chojecki

    A weighted version of the Tang-Zhang sunflower-capacity argument giving the exact logarithmic exponent was developed with GPT-5.4 Pro, using a mass-transport idea from the forum's discussion of problem #1196.

    Identifies the exponent as a variational sunflower-capacity constant, sharpening the Tang-Zhang bounds; the value of that constant itself remains open, as does site acceptance

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