Sharp bounds for uniform union-free hypergraphs
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911679157108736 |
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| author | Liu, Miao Shangguan, Chong Zhang, Chenyang |
| author_facet | Liu, Miao Shangguan, Chong Zhang, Chenyang |
| contents | An $r$-uniform hypergraph is called $t$-union-free if any two distinct subsets of at most $t$ edges have distinct union. The study of union-free hypergraphs has multiple origins and a long history, dating back to the works of Kautz and Singleton (1964) in coding theory, Bollobás and Erdős (1976) in combinatorics, and Hwang and Sós (1987) in group testing. Let $U_t(n,r)$ denote the maximum number of edges in an $n$-vertex $t$-union-free $r$-uniform hypergraph. In this paper, we determine the asymptotic behavior of $U_t(n,r)$, up to a lower order term, for almost all $t\ge 3$ and $r\ge 3$. This significantly advances the understanding of this extremal function, as previously, only the asymptotics of $U_2(n,3)$ and $U_2(n,4)$ were known. As a key ingredient of our proof, we establish the existence of near-optimal locally sparse induced hypergraph packings, which is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_11949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Sharp bounds for uniform union-free hypergraphs Liu, Miao Shangguan, Chong Zhang, Chenyang Combinatorics An $r$-uniform hypergraph is called $t$-union-free if any two distinct subsets of at most $t$ edges have distinct union. The study of union-free hypergraphs has multiple origins and a long history, dating back to the works of Kautz and Singleton (1964) in coding theory, Bollobás and Erdős (1976) in combinatorics, and Hwang and Sós (1987) in group testing. Let $U_t(n,r)$ denote the maximum number of edges in an $n$-vertex $t$-union-free $r$-uniform hypergraph. In this paper, we determine the asymptotic behavior of $U_t(n,r)$, up to a lower order term, for almost all $t\ge 3$ and $r\ge 3$. This significantly advances the understanding of this extremal function, as previously, only the asymptotics of $U_2(n,3)$ and $U_2(n,4)$ were known. As a key ingredient of our proof, we establish the existence of near-optimal locally sparse induced hypergraph packings, which is of independent interest. |
| title | Sharp bounds for uniform union-free hypergraphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.11949 |