Result 177, Combinatorics

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.

Lean formalization Proof

The bigger picture

Why it matters

Can a higher-dimensional network remain strongly connected while each vertex has only a bounded number of neighbors? The manuscript reports constructions showing that a precise parity-based form of this combination is possible in every positive dimension.

What changes?

For each fixed dimension d at least three, the manuscript constructs arbitrarily large finite simplicial complexes: structures built from vertices, edges, triangles and their higher-dimensional counterparts. Vertex degrees stay bounded, and coboundary expansion over the two-element field stays uniformly positive in every degree below d. This expansion measures how reliably parity tests detect face markings far from those generated by lower-dimensional data. The bounds are uniform in size, not asserted uniform across dimensions. Previously known cases cover dimensions one and two.

What does that help mathematicians do?

The claim rules out an apparent obstruction: increasing dimension need not force local connectivity to grow with the size of the complex in order to maintain coboundary expansion. Moreover, in each positive degree below the top dimension, expansion implies that any face marking passing all parity tests comes from lower-dimensional data. Thus these sparse structures have no nontrivial cohomology in those degrees, a precise restriction on their topology.

Are there practical applications?

The immediate value is foundational: researchers gain arbitrarily large sparse models for studying parity-based expansion in any fixed dimension. These models let them investigate strong global constraints without allowing individual vertices ever more neighbors as size grows. The supplied abstract reports an existence construction, not runtime guarantees or a demonstrated practical deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Bounded-degree coboundary expanders in every dimension

September 24, 2026 14 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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's note on what the formalization covers, from lean/docs/177.md.

Bounded-degree coboundary expanders

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The formalization constructs bounded-degree F2\mathbb F_2 coboundary expanders in every dimension d≥3d\ge3. For each such dd, it gives finite pure connected dd-dimensional simplicial complexes with vertex counts tending to infinity, one uniform bound on top-dimensional degree at each vertex, and one positive coboundary-expansion constant in every degree below dd. The constants may depend on dd. The graph and two-dimensional cases used for the paper's all-positive-dimensions conclusion are separate.

Comparator links

Result Comparator statement
Bounded-degree coboundary expanders in dimensions at least three CoboundaryExpanders.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 10 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.