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Ramsey-Style Hypergraph Partition Bound H(n)

Combinatorics · posed by 2019 · solved

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Statement

Let H(n)H(n) be the largest number of vertices in a hypergraph with no isolated vertices and no partition of size greater than nn. With k1=1k_1 = 1 and kn=⌊n/2⌋+k⌊n/2⌋+k⌈n/2⌉k_n = \lfloor n/2 \rfloor + k_{\lfloor n/2 \rfloor} + k_{\lceil n/2 \rceil}, prove H(n)≥c knH(n) \ge c\,k_n for some constant c>1c > 1, already for n=15n = 15, with a constructive algorithm.

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

  1. construction · #1

    GPT-5.4 Pro

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
    ai discovered
    — the result was found by a model.

    GPT-5.4 Pro found a four-way frame construction giving a uniform constant-factor improvement over the known recurrence, starting at n=15n = 15.

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