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arXiv · 2601.18455

Saturation numbers for $3$-uniform Berge-$K_4$

Yihan Chen·Jialin He·Tianying Xie
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Abstract

The saturation number $\text{sat}_r(n,\mathcal{F})$ is the minimum number of hyperedges in an $r$-uniform $\mathcal{F}$-saturated hypergraph on $n$ vertices. We determine this parameter for $3$-uniform Berge-$K_4$ hypergraphs, proving that $\text{sat}_3(n,\text{Berge-}K_4)=n$ for $n =5,7,8$ and $n\ge 96$, while $\text{sat}_3(6,\text{Berge-}K_4)=5$. This resolves a problem posed by English, Kritschgau, Nahvi, and Sprangel~\cite{EKNS2024} for large $n.$ Using a computer search, we classify all extremal hypergraphs for $5\le n\le 8.$ For $n\geq 96$, we further show the existence of many non-isomorphic extremal families. Our approach synthesizes structural insights with computational power.

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Yihan Chen, Jialin He, Tianying Xie. 2026-01-26. Saturation numbers for $3$-uniform Berge-$K_4$. https://arxiv.org/abs/2601.18455

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