Abundance of Unique Subhypergraphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916075477663744 |
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| author | Shu, Xichao Wu, Zhuo Xue, Yisai |
| author_facet | Shu, Xichao Wu, Zhuo Xue, Yisai |
| contents | Given $k$-uniform hypergraphs $G$ and $H$, we say that $G$ is a unique subhypergraph of $H$ if $H$ contains exactly one subhypergraph isomorphic to $G$. For an $n$-vertex $k$-graph $H$, let $f_k(H)$ be the number of non-isomorphic unique subhypergraphs of $H$, normalized by $2^{\binom n k}/n!$, and let $f_k(n)$ be the maximum of $f_k(H)$ over all $n$-vertex $k$-graphs $H$. In the graph case $k=2$, Erdős asked whether there exists a constant $δ>0$ such that $f_2(n)>δ$ for all $n$, offering \$100 for a proof and \$25 for a disproof.
Recently, Bradač and Christoph answered this question in the negative,, proving that $f_2(n)$ tends to $0$, or equivalently that no $n$-vertex graph contains a positive proportion of all $n$-vertex graphs as unique subgraphs. In this paper we show that the situation is fundamentally different for $k$-uniform hypergraphs with $k\ge3$. In particular, for every fixed integer $k\ge 3$, we prove that $\liminf_{n\to\infty} f_k(n) \ge 2/9$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_02546 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Abundance of Unique Subhypergraphs Shu, Xichao Wu, Zhuo Xue, Yisai Combinatorics Given $k$-uniform hypergraphs $G$ and $H$, we say that $G$ is a unique subhypergraph of $H$ if $H$ contains exactly one subhypergraph isomorphic to $G$. For an $n$-vertex $k$-graph $H$, let $f_k(H)$ be the number of non-isomorphic unique subhypergraphs of $H$, normalized by $2^{\binom n k}/n!$, and let $f_k(n)$ be the maximum of $f_k(H)$ over all $n$-vertex $k$-graphs $H$. In the graph case $k=2$, Erdős asked whether there exists a constant $δ>0$ such that $f_2(n)>δ$ for all $n$, offering \$100 for a proof and \$25 for a disproof. Recently, Bradač and Christoph answered this question in the negative,, proving that $f_2(n)$ tends to $0$, or equivalently that no $n$-vertex graph contains a positive proportion of all $n$-vertex graphs as unique subgraphs. In this paper we show that the situation is fundamentally different for $k$-uniform hypergraphs with $k\ge3$. In particular, for every fixed integer $k\ge 3$, we prove that $\liminf_{n\to\infty} f_k(n) \ge 2/9$. |
| title | Abundance of Unique Subhypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2606.02546 |