Ramsey-Style Hypergraph Partition Bound H(n)

Precise statement

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

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT-5.4 Pro

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

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Verified by the problem's contributor, who is preparing the argument for publication; problem and status tracked publicly.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-03-24

   Epoch AI open problem: a Ramsey-style problem on hypergraphs

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

construction (source-reported)

Source-reported tools: construction.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.