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Combinatorics · Result 177

Bounded-degree coboundary expanders

1 paperLean

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.

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▶Introduction: the problem and why it mattersin preparation
▶The presented solutionin preparation

Papers 1

First page of Bounded-degree coboundary expanders in every dimension

Bounded-degree coboundary expanders in every dimension

OpenAI · September 24, 2026

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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@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}
}

What was formalized in Lean

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

Scope

The formalization constructs bounded-degree \(\mathbb F_2\) coboundary expanders in every dimension \(d\ge3\). For each such \(d\), it gives finite pure connected \(d\)-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 \(d\). The constants may depend on \(d\). The graph and two-dimensional cases used for the paper's all-positive-dimensions conclusion are separate.

Comparator links

ResultComparator statement
Bounded-degree coboundary expanders in dimensions at least threeCoboundaryExpanders.lean

Scope note on GitHub →