Towards the Erdős matching conjecture for 4-uniform hypergraphs: stability and applications
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915811293134848 |
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| author | Frankl, Peter Lu, Hongliang Ma, Jie Wu, Yuze |
| author_facet | Frankl, Peter Lu, Hongliang Ma, Jie Wu, Yuze |
| contents | A famous conjecture of Erdős asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for $k=3$ due to the first author. In this paper, we make progress towards the $4$-uniform case, proving the conjecture for $n\ge 5s$ and sufficiently large $n$, thereby taking a first step analogous to the $3$-uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum $d$-degree threshold for matchings in $5$- and $6$-uniform hypergraphs, in a strengthened form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_19230 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Towards the Erdős matching conjecture for 4-uniform hypergraphs: stability and applications Frankl, Peter Lu, Hongliang Ma, Jie Wu, Yuze Combinatorics A famous conjecture of Erdős asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem has been central in extremal combinatorics, with substantial progress in the literature, including a complete solution for $k=3$ due to the first author. In this paper, we make progress towards the $4$-uniform case, proving the conjecture for $n\ge 5s$ and sufficiently large $n$, thereby taking a first step analogous to the $3$-uniform case. The main technical contribution is a stability result of independent interest. We further apply this stability to resolve two new instances of conjectures on the minimum $d$-degree threshold for matchings in $5$- and $6$-uniform hypergraphs, in a strengthened form. |
| title | Towards the Erdős matching conjecture for 4-uniform hypergraphs: stability and applications |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2602.19230 |