1 Β· The essentials
The manuscript describes how to build arbitrarily large, finite shapes from simple pieces, such as points, edges, triangles, and their higher-dimensional counterparts. It does this for every dimension from three upward. As the shapes grow, the number of connections at any point stays below a fixed limit. At the same time, the shapes retain a mathematical property called coboundary expansion, measured using arithmetic with just zero and one. For each chosen dimension, the guarantees remain uniform as the shapes grow and apply at every level below the top dimension. Combined with the already known one- and two-dimensional cases, this gives existence in every positive dimension. The main result has a Lean formalization, although the supplied description does not specify its exact scope.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine building larger and larger assemblies from triangular panels and higher-dimensional building blocks while limiting how many connections meet at each corner. The result says that this local connection limit can coexist with a uniform mathematical expansion condition throughout the assembly; the physical picture does not capture that condition itself.
Why it matters
The result combines arbitrarily large size with two uniform requirements: bounded connections at each vertex and coboundary expansion below the top dimension. The new constructions cover every dimension at least three, completing the existence statement when joined with the known lower-dimensional cases.
2 Β· Going further
The precise problem is to construct, for each integer , arbitrarily large finite -dimensional simplicial complexes whose vertex degrees are uniformly bounded and whose coboundary expansion is uniform in every degree below . Here a simplicial complex is a structure assembled from vertices, edges, triangles, and higher-dimensional simplices, with their faces included. The coefficient field uses arithmetic modulo two.
The key combination is that increasing the size does not require giving up either the vertex-degree bound or the expansion guarantee. For each fixed , the statement provides uniform bounds across arbitrarily large complexes. It does not state that the same bounds work independently of dimension, nor does it assert expansion in the top degree .
The abstract describes the contribution as a construction, but supplies no details of its mechanism or argument. Its stated context is that the graph and two-dimensional cases were already known; combining those cases with the construction yields existence in every positive dimension. Only the main manuscript is supplied, so no relationship to companion manuscripts can be described. The main result has a Lean formalization, but the supplied material does not identify which additional claims, if any, that formalization covers.
Useful notions
- Vertices, edges, and graphs
- Triangles and higher-dimensional simplices
- Arithmetic modulo two
- Uniform bounds across a growing family
What was known before / what is new
Before
The source states that the graph case and the two-dimensional case were already known. It gives no further historical details or account of those constructions.
New
The manuscript presents constructions in every integer dimension . These are arbitrarily large finite complexes with uniformly bounded vertex degrees and uniform coboundary expansion in every degree below . Together with the known lower-dimensional cases, this establishes the stated existence result in every positive dimension.
Limits and assumptions
The new construction concerns finite simplicial complexes of each fixed integer dimension , with coefficients in and expansion only in degrees below . Uniformity is asserted as size grows; dimension-independent bounds are not stated. The main result has a Lean formalization, whose exact scope is not specified in the supplied material. The abstract gives neither quantitative bounds nor construction details, and it identifies no remaining open problem.
Mini glossary
- Simplicial complex β
- A structure made of vertices, edges, triangles, and higher-dimensional simplices, including the faces of each piece.
- Vertex degree β
- The number of edges meeting a vertex.
- Uniform bound β
- A bound that remains fixed across the growing family under consideration.
- β
- The field with elements zero and one, with addition and multiplication performed modulo two.
- Lean formalization β
- A mathematical statement and its verification expressed in the Lean proof assistant.
Keywords : Simplicial complexes Β· Bounded vertex degree Β· Coboundary expansion
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 Β· Technical details and sources
Claim made in the manuscript (source text)
Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d β₯ 3, with uniformly bounded vertex degrees and uniform π½2 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.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
Bounded-degree coboundary expanders in every dimension
24/09/2026For every integer d β₯ 3, we construct arbitrarily large finite d-dimensional simplicial complexes with uniformly bounded vertex degrees and uniform π½2 coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of bounded-degree π½2 coboundary expanders in every positive dimension.