The Erdős matching conjecture

Determine the largest r-uniform hypergraph on n vertices with no k disjoint edges; Erdős conjectured one of two simple families wins.

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1	# The Erdős matching conjecture
2
3	How many $r$-element sets can one choose from an $n$-element ground set without ever finding $k$ pairwise disjoint ones? Erdős conjectured in 1965 that one of two simple families is always optimal: all $r$-sets inside a set of $rk-1$ points, or all $r$-sets that meet a fixed set of $k-1$ points. The conjecture contains the Erdős–Ko–Rado theorem and the Erdős–Gallai theorem as special cases. It has been proved for $r=3$, and an October 2026 preprint proves it whenever $k$ is large compared with $r$, but it is open in general.
4
5	## Problem
6
7	Let $f(n;r,k)$ be the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices that contains no $k$ pairwise disjoint edges. Is it true that for all $r\ge3$ and $n\ge rk$
8
9	$$
10	f(n;r,k)=\max\left\{\binom{rk-1}{r},\ \binom{n}{r}-\binom{n-k+1}{r}\right\}?
11	$$
12
13	The first term counts all $r$-sets of a fixed set of $rk-1$ vertices; the second counts all $r$-sets meeting a fixed set of $k-1$ vertices. Neither family contains $k$ disjoint edges, so the right-hand side is always a lower bound. (For $n<rk$ there cannot be $k$ disjoint edges at all.)
14
15	## Background
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17	Erdős posed the conjecture in 1965 and proved it when $n$ is large in terms of $r$ and $k$. For $r=2$ it is a theorem of Erdős and Gallai (1959), and for $k=2$ it reduces to the Erdős–Ko–Rado theorem on intersecting families. The second construction is the larger one once $n\ge(r+1)k$, so the hard range lies where the two constructions are close.
18
19	## What is known
20
21	- Large $n$. After Erdős's original result, the range was improved by Bollobás, Daykin and Erdős ($n\ge 2kr^3$, 1976), Huang, Loh and Sudakov ($n\ge 3kr^2$, 2012), Frankl, Łuczak and Mieczkowska ($n>2kr^2/\log r$, 2012) and Frankl (2013). Cao, Liu and Zhang (2026, preprint) showed that, for $k$ large in terms of $r$, the conjecture holds whenever $n\ge(r+1)(k-1)$, close to where the two constructions trade places.
22	- Large matchings. Chi, Sun and Wang (October 2026, preprint) proved the conjecture for every fixed $r\ge3$, every sufficiently large $k$ and every $n\ge rk$, by settling a conjecture of Łuczak, Mieczkowska and Šileikis on tail probabilities of sums of independent random variables. The open cases are now those in which $k$ is small compared with $r$.
23	- Small $n$. Kleitman (1968) settled $n=kr$; Frankl (2017) and Kolupaev and Kupavskii (2023) extended this to ranges just above $kr$.
24	- Small uniformity. The case $r=3$ has been solved completely by Frankl. For $r=4$, Frankl, Lu, Ma and Wu (2026, preprint) proved the conjecture for $n\ge5(k-1)$ and $n$ sufficiently large. A manuscript posted in September 2026 (O. Babanskyy, listed on the problem's proof-claims page) claims the whole case $r=4$, with computer-checked finite cases; it has not been independently verified.
25	- Claims. A complete proof posted in February 2026 ([arXiv:2602.01471](https://arxiv.org/abs/2602.01471)) was found to contain an error on the problem's forum and was withdrawn by its author in June 2026.
26
27	## What would count as progress
28
29	- A verified proof of the case $r=4$ for all $n$ and $k$, or of the conjecture for small $k$ relative to $r$ in a new range of $n$.
30	- A counterexample: an explicit $r$-uniform hypergraph with no $k$ disjoint edges and more edges than the conjectured maximum, which can be checked by computing its matching number.
31	- Exact values of $f(n;r,k)$ for small parameters, with certificates.
32	- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1020.lean).
33
34	## References
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36	1. P. Erdős and T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959), 337–356. [doi:10.1007/BF02024498](https://doi.org/10.1007/BF02024498)
37	2. H. Huang, P.-S. Loh and B. Sudakov, The size of a hypergraph and its matching number, Combin. Probab. Comput. 21 (2012), 442–450. [doi:10.1017/S096354831100068X](https://doi.org/10.1017/S096354831100068X)
38	3. P. Frankl, Improved bounds for Erdős' matching conjecture, J. Combin. Theory Ser. A 120 (2013), 1068–1072. [doi:10.1016/j.jcta.2013.01.008](https://doi.org/10.1016/j.jcta.2013.01.008)
39	4. P. Frankl, H. Lu, J. Ma and Y. Wu, Towards the Erdős matching conjecture for 4-uniform hypergraphs: stability and applications (2026). [arXiv:2602.19230](https://arxiv.org/abs/2602.19230)
40	5. M. Cao, H. Liu and H. Zhang, A near-optimal linear range for the Erdős matching conjecture (2026). [arXiv:2608.19118](https://arxiv.org/abs/2608.19118)
41	6. C. Chi, S. Sun and Y. Wang, A proof of the Erdős matching conjecture for large matchings via sharp tail probability for sums of i.i.d. nonnegative random variables (2026). [arXiv:2610.16203](https://arxiv.org/abs/2610.16203)
42
43	## Source
44
45	Imported from [Erdős Problems 1020](https://www.erdosproblems.com/1020), curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.