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A computer-assisted proof of the four-uniform Erdős matching conjecture

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatorics

Contributed by Oleksiy Babanskyy ↗

Version 1 / Oct 29, 2026 / CC BY 4.0

Abstract

We prove the Erdős matching conjecture for four-uniform hypergraphs for every matching parameter and every admissible number of vertices. The proof combines the finite-board method of Hou, Hu and Liu with ordered local configurations, exact integer counts and certified inequalities. Small matching parameters are treated on full labelled configurations. Two fixed counting inequalities and two endpoint anchors propagate polynomially across the remaining range; 51 endpoint records and the two fixed inequalities suffice for the intermediate cardinality inputs. Dense triple supports admit direct slab and majority proofs. A classical shadow bound and a deletion argument complete the passage from these local bounds to the global theorem. The strict margins also yield clique stability with explicit constants for shifted families whose matching number is preserved on deleting the first vertex, and classifications of extremizers in specified ranges. The finite parts are supported by exact certificates and exhaustive computations with stated mathematical specifications.

Provenance statement

The author directed the research programme and selected its scope. AI systems contributed substantially to mathematical exploration, proposed lemmas and counterchecks, certificate and verification code, manuscript drafting and revision, and adversarial reviews. The author reports using GPT-5.6 Sol, Astra, GPT-5 Pro and GPT-6 Pro via ChatGPT. These are author-reported names; the manuscript discloses extensive AI assistance but does not identify model versions. Individual proof steps are not attributed to particular models. The proof develops the finite-board method of Hou, Hu and Liu. Eight exhaustive computations from their work are retained as credited dependencies and are rerun from a separately obtained, hash-pinned source archive. The repository provides exact certificates, verification programs and reproduction instructions. Verification to date consists of AI-assisted reviews and author-side computational replays. The mathematical proof has not been checked and understood by a human, and no end-to-end formal proof is supplied. Oleksiy Babanskyy is responsible for the manuscript and its claims. The claimed theorem covers every matching parameter in uniformity four. It does not solve the conjecture in arbitrary uniformity; the additional stability and equality results have the restricted domains stated in the manuscript. The same manuscript is available as aiXiv 2609.03278 v1: https://aixiv.online/abs/2609.03278v1 Source, certificates and reproduction instructions: https://github.com/aconsciousfractal/Four-Uniform-Erdos-Matching-Conjecture/tree/e0904541b91b0246d843f5845f638733247fbdba

Tools used

OpenAI
ChatGPTVersion GPT-5.6 Sol; Astra; GPT-5 Pro; GPT-6 Pro
Python Software Foundation
Python
Zig Software Foundation
ZigVersion 0.16.0
OpenJS Foundation
Node.js

References

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Show all 16 referencesShow fewer references
  1. M. Cao, H. Liu and H. Zhang, A Near-Optimal Linear Range for the Erdős Matching Conjecture , arXiv:2608.19118v2 (https://arxiv.org/abs/2608.19118v2). Theorems 1.2 and 1.4; Remark 1.3 and Section 6.1.arXiv
  2. J. Hou, C. Hu and X. Liu, A finite-board reduction for the Erdős Matching Conjecture and the 4-uniform case via exact certificates , arXiv:2605.26060v1 (https://arxiv.org/abs/2605.26060v1). Sections 2 and 4 for the attributed trace and selection mechanisms; Lemmas 5.1--5.3, specified C1/C2 parts of Theorem 9.2 and Appendix A.1 for the finite ingredients reconstructed in Sections app:global -- app:range92 . Lemma 3.1 is not an input to the present global proof.arXiv
  3. P. Frankl, On the maximum number of edges in a hypergraph with given matching number , Discrete Appl. Math. 216 (2017), 562--581. doi:10.1016/j.dam.2016.08.003 (https://doi.org/10.1016/j.dam.2016.08.003). Theorem locators for the deletion inequality and the historical announcement follow the later author copy (https://www.renyi.hu/~pfrankl/2016-2.pdf), Proposition 5.1 and Theorem 13.2; they are distinct from the first arXiv version 1205.6847v1 (https://arxiv.org/abs/1205.6847v1).DOI
  4. T. Łuczak, K. Mieczkowska and M. Š ileikis, On maximal tail probability of sums of nonnegative, independent and identically distributed random variables , Statistics & Probability Letters 129 (2017), 12--16, doi:10.1016/j.spl.2017.04.024 (https://doi.org/10.1016/j.spl.2017.04.024); consulted version arXiv:1602.03547v1 (https://arxiv.org/abs/1602.03547v1). Conjecture 1.1 and Lemma 3.2.arXivDOI
  5. SciNet, Erdős matching conjecture (#1020) confirmed by exact computation in five complete open windows , public dossier, 27 July 2026. Finding and finite-value table (https://api.scinet.pub/f/aec4527d-0276-45b3-82ab-98aa3f965d75); repository at commit e36280d1 (https://github.com/scinet-ai/math-number-theory/blob/e36280d1d511422e9447ae98f8c49bcb644fd678/erdos-1020/README.md). Compared as prior computational work; its solvers are not proof inputs.link
  6. T. K. Mishra, Erdős Matching (Conjecture) Theorem , arXiv:2602.01471v6 (https://arxiv.org/abs/2602.01471v6), withdrawn 1 June 2026. Cited only for the withdrawal status of the earlier complete-solution claim, not as a theorem input.arXiv

Version history

  1. v1Submitted by Oleksiy BabanskyyInitial depositCurrentOct 28, 2026