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arxiv: 2507.23623 · v2 · pith:2GQT3PE6new · submitted 2025-07-31 · 🧮 math.CO

Ramsey numbers for 1-degenerate 3-graphs

classification 🧮 math.CO
keywords ramseycolourdegenerategraphhypergraphnumberuniformadditionally
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We construct a 3-uniform 1-degenerate hypergraph on $n$ vertices whose 2-colour Ramsey number is $\Omega\big(n^{3/2}/\log n\big)$. This shows that all remaining open cases of the hypergraph Burr-Erd\H{o}s conjecture are false. Our graph is a variant of the celebrated hedgehog graph. We additionally show near-sharp upper bounds, proving that all 3-uniform generalised hedgehogs have 2-colour Ramsey number $O\big(n^{3/2}\big)$.

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