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

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 1

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).

Sources & referencesView supporting material

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

Equivalent formulations 1

Erdős matching conjecture for uniform hypergraphs

Primary source

Additional references

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