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Bounded-degree coboundary expanders in every dimension
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Abstract
For every integer d ≥ 3, we construct arbitrarily large finite d-dimensional simplicial complexes with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of bounded-degree 𝔽2 coboundary expanders in every positive dimension.
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Notes
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Formalization
- OpenAI math release, Lean scope notes for family 177: Bounded-degree coboundary expanders (lean/docs/177.md). Describes the formalization's scope: for every d ≥ 3, pure connected d-complexes with growing vertex counts, uniformly bounded vertex degree and uniform 𝔽₂ coboundary expansion in all degrees below d; the graph and two-dimensional cases are not part of it. [family-level]
- OpenAI math release, Comparator statement CoboundaryExpanders.lean (openai/math, Lean 4/Mathlib). Self-contained Lean statement
MainStatementof Theorem 1.1, covering weighted norms, the coboundary map and distance to coboundaries; in this challenge file the theorem itself is left assorry. - OpenAI math release, Main.lean, Combinatorics/Coboundary (openai/math, commit adc7f12). Contains the theorem
main : MainStatement, described in its docstring as the manuscript's full theorem; the proof fixes a large odd prime and reduces the statement to group-homology vanishing inputs.
Related prior literature mentioned by others
- Izhar Oppenheim and Inga Valentiner-Branth, New cosystolic high-dimensional expanders from KMS groups (arXiv:2504.05823v2, 9 September 2026). The AI-written Open Mathematics Courses exposition identifies this work as the expansion input used alongside the manuscript's cohomology-vanishing argument.
Possibly relevant literature (found by the agent, not checked)
- M. Hopkins and A. Ray, A Simple Sub-Polynomial Degree Coboundary Expander (arXiv:2605.22173, 2026-05-21). A recent combinatorial coboundary-expander construction with sub-polynomial rather than bounded degree, relevant to the paper's account of the state of the art before Theorem 1.1.
- Y. Dikstein, S. Liu and A. Wigderson, Sparser Abelian High Dimensional Expanders (arXiv:2411.08839, 2024-11-13). Gives a two-dimensional complex that is both a spectral and a coboundary expander, with unbounded degree, relevant to the paper's "Prior work" comparison of degree conditions.
- R. O'Donnell and N. G. Singer, Coboundary expansion inside Chevalley coset complex HDXs (arXiv:2411.05916, 2024-11-08). Proves 1-coboundary expansion in the links of B₃ coset complexes over 𝔽_{5^k}[x] using unipotent-group cohomology vanishing, which parallels the paper's coset-complex and cohomology-vanishing approach.
Digests and expositions
- Latent Seminar, Bounded-degree coboundary expansion (7 October 2026). Explains the distinction between cocycles and coboundaries, the congruence-tower vanishing strategy and the skeleton-weight comparison, while explicitly disclaiming complete proof verification or a Lean audit.
- Open Mathematics Courses, Coboundary expanders of bounded degree (undated). Three lessons develop weighted coboundary expansion and the polynomial-matrix coset construction, deriving the headline theorem conditionally on two explicitly unproved inputs: expansion relative to cocycles and cohomology vanishing. [AI-generated per source]
News and blogs
- Gil Kalai, Updates: Sharing AI progress on mathematics (amazing!); and my lecture plans (Combinatorics and more, 2026-10-07). A section on Ramanujan graphs and higher-dimensional expanders mentions this paper's bounded-degree 𝔽₂ coboundary expanders for d ≥ 3 and links the result to Kalai's earlier introductory posts on high-dimensional expanders.
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 177
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026,
author = {{OpenAI}},
title = {{Bounded-degree coboundary expanders in every dimension}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026/paper.pdf}{OAI:Bounded-degree-coboundary-expanders-in-every-dimension-September-24-2026}},
year = {2026}
}Lean formalization
OpenAI provides Lean files for this family: lean/docs/177.md. The Lean statements have not been checked by people against the paper's claims.
Family 177: Bounded-degree coboundary expanders
Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽2 coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension.
This is the only paper in this family.