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Babai and Frankl's Oddtown Question for Composite Moduli

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babai-frankl-oddtown-compositeCombinatoricsposed by László Babai, Péter Frankl, 1992recorded: disproved

1 attempt · no person has looked

Statement

An ℓ\ell-Oddtown is a family of subsets of an nn-element set whose set sizes are not divisible by ℓ\ell while all pairwise intersection sizes are. Berlekamp and Graver showed the maximum size is nn for prime ℓ\ell, Babai and Frankl extended this to prime powers and asked whether nn still holds for other moduli, a question open even for ℓ=6\ell = 6. Bukh, Chao and Zheng answer it negatively with an explicit superlinear construction, complemented by new upper bounds.

Context

A named question from Babai and Frankl's widely used linear-algebra-methods text, open for decades and known across extremal combinatorics.

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1 attempt

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  • #1

    Attempt 1

    constructionGPT-5.6 Sol with Boris Bukh, Ting-Wei Chao, Zeyu Zheng ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    GPT-5.6 Sol
    people
    Boris Bukh, Ting-Wei Chao, Zeyu Zheng

    The lower-bound construction in Section 2 was first proposed by GPT-5.6 Sol in response to prompts from the authors; the upper-bound results were obtained without AI assistance. (The disclosure was added in the paper's second version.)

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