Let M_s^r be a matching of size s in an r-uniform hypergraph, and let operatornameex_r(n,M_s^r) denote the maximum number of edges in an n-vertex r-uniform hypergraph containing no copy of M_s^r. Erdős Matching Conjecture. For integers s,r≥ 2 and n≥ sr-1, operatornameex_r(n,M_s^r)≤ maxbinomsr-1r,binomnr-binomn-s+1r. This conjecture extends the Erdős–Gallai theorem from graphs to uniform hypergraphs and predicts the extremal number for forbidding a matching of size s. The paper describes its main theorem as a spectral confirmation in the sufficiently large-n regime, while the full extremal conjecture is not stated as completely resolved here.

Erdős Matching Conjecture for uniform hypergraphs

At least 7 years old · documented by 2018

Let $M_s^r$ be a matching of size $s$ in an $r$ -uniform hypergraph, and let $\operatorname{ex}_r(n,M_s^r)$ denote the maximum number of edges in an $n$ -vertex $r$ -uniform hypergraph containing no copy of $M_s^r$ . Erdős Matching Conjecture. For integers $s,r\geq 2$ and $n\geq sr-1$ ,

\operatorname{ex}_r(n,M_s^r)\leq \max\left\{\binom{sr-1}{r},\binom{n}{r}-\binom{n-s+1}{r}\right\}.

This conjecture extends the Erdős–Gallai theorem from graphs to uniform hypergraphs and predicts the extremal number for forbidding a matching of size $s$ . The paper describes its main theorem as a spectral confirmation in the sufficiently large- $n$ regime, while the full extremal conjecture is not stated as completely resolved here.

Equivalent formulations 1Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Erdős matching conjecture for uniform hypergraphs
Let $k\text{-graph}$ mean a $k$ -uniform hypergraph, let $\beta$ be a positive integer, and let $u(H)$ denote the maximum size of a matching in a $k$ -graph $H$ . Suppose that $k\geq 2$ , $\beta\geq 1$ , and $n\geq k(\beta+1)-1$ are integers, and that $H$ is an $n$ -vertex $k$ -graph with $\mu(H)\leq\beta$ .

Erdős matching conjecture. The number of edges of $H$ satisfies
$|E(H)|\leq\max\left\{\binom{k(\beta+1)-1}{k},\binom{n}{k}-\binom{n-\beta}{k}\right\}.$
The conjecture is a central extremal problem for uniform hypergraphs with bounded matching number. It remains open in full generality, although the paper obtains a spectral version for sufficiently large $n$ .
source: Yi Xu and Yi-Zheng Fan, “Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number”, arXiv:2607.23560 (2026).

References

Primary source

Liying Kang, Yongchun Lu, Xiying Yuan and Junpeng Zhou, “A Spectral Confirmation of the Erdős Matching Conjecture”, arXiv:2607.07392 (2026).

Additional references

14 papers in this index state this conjecture (2018–2026). The statement above is taken from the most recent of them; the others are arXiv:2606.24529, arXiv:2604.19183, arXiv:2603.06415, arXiv:2511.21628, arXiv:2511.17000, arXiv:2508.17683, arXiv:2504.14389, arXiv:2404.09720, arXiv:2403.04289, arXiv:2012.15142, arXiv:2011.14252, arXiv:2002.06601, and 1 more.

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Erdős Matching Conjecture for uniform hypergraphs

Erdős matching conjecture for uniform hypergraphs

Primary source

Additional references

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