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The Erdős Matching Conjecture for 4-uniform hypergraphs

  • Jianfeng Hou
  • Caiyun Hu
  • Xizhi Liu
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Abstract

We prove the Erdős Matching Conjecture for $4$-uniform hypergraphs for every matching number s≥6004. Specifically, for integers s≥ 6004 and n≥ 4s+4, if a n-vertex 4-uniform hypergraph ℱ does not contain a matching of size s+1, then |ℱ|≤ max{binom4s+34,binom n4−binomn−s4}. Our proof introduces a finite-board reduction for general uniformity. It reduces the global conjecture to a lower-uniformity bound and a fixed finite optimization at the two adjacent vertex numbers where the candidate constructions exchange dominance. In the $4$-uniform case the board has $19$ vertices, and its weighted inequality splits into $19$-, $15$-, and $11$-vertex layers. The hardest layer is resolved by exact rational dual certificates and deterministic integer searches. All computer-assisted steps are checked in exact arithmetic by verifiers that reconstruct the finite systems directly from their mathematical definitions.

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Cite

BibTeXRIS
BibTeX
@misc{hou2026erdős,
  title = {{The Erdős Matching Conjecture for 4-uniform hypergraphs}},
  author = {Jianfeng Hou and Caiyun Hu and Xizhi Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.26060},
  eprint = {2605.26060},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.26060},
}
RIS
TY  - UNPB
TI  - The Erdős Matching Conjecture for 4-uniform hypergraphs
AU  - Jianfeng Hou
AU  - Caiyun Hu
AU  - Xizhi Liu
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2605.26060
UR  - https://arxiv.org/abs/2605.26060
AB  - We prove the Erdős Matching Conjecture for $4$-uniform hypergraphs for every matching number s≥6004. Specifically, for integers s≥ 6004 and n≥ 4s+4, if a n-vertex 4-uniform hypergraph ℱ does not contain a matching of size s+1, then |ℱ|≤ max{binom4s+34,binom n4−binomn−s4}. Our proof introduces a finite-board reduction for general uniformity. It reduces the global conjecture to a lower-uniformity bound and a fixed finite optimization at the two adjacent vertex numbers where the candidate constructions exchange dominance. In the $4$-uniform case the board has $19$ vertices, and its weighted inequality splits into $19$-, $15$-, and $11$-vertex layers. The hardest layer is resolved by exact rational dual certificates and deterministic integer searches. All computer-assisted steps are checked in exact arithmetic by verifiers that reconstruct the finite systems directly from their mathematical definitions.
ER  -