# P2684: Largest first-homology torsion from forty-five triangles on eleven vertices

- ID: `P2684`
- Reference: `complete-skeleton-eleven-torsion`
- Page: https://theoremdb.org/statements/P2684
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(K\) contain the complete graph on vertices \(\{0,\ldots,10\}\) and exactly \(45\) triangular faces. Among choices for which \(H_1(K;\mathbb Z)\) is finite, determine the largest possible order of \(H_1(K;\mathbb Z)\).

### Remarks

- **Remark.** There are C(11,3)=165 possible triangular faces.
- **Remark.** The cycle rank of the complete one-skeleton is 45, so 45 face boundaries can kill all rational first homology and leave a finite torsion group.

### What counts as a solution

- Give 45 faces attaining the maximum and a determinant or Smith-form certificate, together with an exhaustive or certified determinant bound for every other selection.

## Status

An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18. [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (The certified interval is 74 through 387,420,489).** An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18.

Let \(M(K)\) be the maximum in the question. The exact replay in this record constructs a 45-face complex with
\[
H_1(K;\mathbb Z)\cong\mathbb Z/74\mathbb Z,
\]
so \(M(K)\ge74\).

Kalai's torsion theorem states that the torsion part of \(H_{d-1}\) for a \(d\)-complex on \(n\) vertices has order at most
\[
(\sqrt{d+1})^{\binom{n-2}{d}}.
\]
At \((n,d)=(11,2)\), this is
\[
(\sqrt3)^{\binom92}=3^{18}=387{,}420{,}489.
\]
Consequently
\[
\boxed{74\le M(K)\le387{,}420{,}489}.
\]
The construction improves the candidate record's lower bound of 51. The search used determinant-preserving basis exchanges followed by exact integer checks. It did not enumerate all \(\binom{165}{45}\) face sets, so the exact maximum remains open.

### Background and intake notes

The current certified interval for the maximum torsion order is 51 through 54353589638.

- Original intake status: Novelty remains unverified. No primary-source status audit was completed for this exact finite complex.
- Delete the star edges at vertex 0 to obtain the standard 45-element cycle basis. Each triangle boundary is then a column with one or three nonzero entries.
- For any full-rank selection, the absolute determinant of the 45 by 45 boundary matrix is exactly the torsion order. Smith normal form should be retained for every record construction.
- Hadamard gives the coarse upper bound floor(3^(45/2))=54353589638. Determinant maximization needs stronger minor and matroid bounds.

- Recorded example: A 45-face complex with torsion order 51 is recorded in the computation field.

### Other known results

- **Computation 1** (reproduced): The reduced boundary matrix has determinant 74 and Smith diagonal 1 repeated 44 times followed by 74. [2](#reference-2)

### Prior approaches

- **Route 1** (supported): Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated. [3](#reference-3) [4](#reference-4) [2](#reference-2) [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): Standard-library Python reconstructs the 45 by 45 boundary matrix and certifies the Smith diagonal with two exact Bareiss determinants.

### Computational notes

- Ten thousand seeded determinant-exchange searches produced a full-rank boundary matrix of determinant -51. Its faces are 359,038,127,017,68A,57A,048,016,569,058,13A,578,129,67A,349,025,237,09A,45A,047,159,178,679,28A,568,146,89A,479,148,238,36A,26A,49A,789,036,245,246,15A,249,134,235,37A,02A,039,01A, where A denotes vertex 10 and each three-character word is a face. Fraction-free elimination independently replayed the determinant.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `complete-skeleton-eleven-torsion`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Art M. Duval, Caroline J. Klivans, and Jeremy L. Martin, “Simplicial matrix-tree theorems”. Trans. Amer. Math. Soc. 361 (2009), no. 11, 6073-6114. arXiv:0802.2576 (2008). Torsion-weighted cellular spanning-tree framework https://arxiv.org/abs/0802.2576
   - preprint; reference source; arXiv:0802.2576, version checked 2026-07-25; checked 2026-07-25
   - Source use: citation_only
   - For Largest first-homology torsion from forty-five triangles on eleven vertices: The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.
2. <a id="reference-2"></a>Andrew Newman, “Small simplicial complexes with prescribed torsion in homology”. arXiv:1707.09271 (2017). Andrew Newman, Small simplicial complexes with prescribed torsion in homology, Theorem 4 part 1 as quoted from Kalai; lower construction and Smith certificate in this record; Theorem 4 part 1 and concluding Question 3 https://arxiv.org/abs/1707.09271
   - Also cited at Theorem 4 part 1 and concluding Question 3
   - Also cited at Andrew Newman, Small simplicial complexes with prescribed torsion in homology, Theorem 4 part 1 as quoted from Kalai; lower construction and Smith certificate in this record
   - Also cited at Exact integer certificate in cset11-artifact-boundary-smith-replay, produced 2026-07-25
   - preprint; reference source; arXiv:1707.09271, version checked 2026-07-25; checked 2026-07-25
   - Source use: citation_only
   - For Largest first-homology torsion from forty-five triangles on eleven vertices: The certified interval is 74 through 387,420,489. An explicit rational 2-hypertree has cyclic first homology of order 74, while Kalai's general torsion theorem gives the upper bound 3^18. The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.
   - Source named by the research packet.
3. <a id="reference-3"></a>Nati Linial and Yuval Peled, “Enumeration and randomized constructions of hypertrees”. arXiv:1801.02423 (2018). Nati Linial and Yuval Peled, Enumeration and randomized constructions of hypertrees, Definition 1.1 and Theorem 1.2; Andrew Newman, Small simplicial complexes with prescribed torsion in homology, Theorem 4 part 1 and Question 3; Definition 1.1, boundary-matrix characterization, and Theorem 1.2 https://arxiv.org/abs/1801.02423
   - Also cited at Definition 1.1, boundary-matrix characterization, and Theorem 1.2
   - preprint; reference source; arXiv:1801.02423, version checked 2026-07-25; checked 2026-07-25
   - Source use: citation_only
   - For Largest first-homology torsion from forty-five triangles on eleven vertices: The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.
4. <a id="reference-4"></a>Sabine Grabner, “Seslerio-caricetum sempervirentis andCaricetum ferrugineae in the Northern Calcareous Alps”. Folia Geobotanica et Phytotaxonomica 32(3) (1997), 297-311. DOI 10.1007/BF02804009. Israel Journal of Mathematics 45 (1983), 337-351 https://doi.org/10.1007/BF02804009
   - scholarly_publication; reference source; version of record; checked 2026-08-01
   - Source use: citation_only
   - For Largest first-homology torsion from forty-five triangles on eleven vertices: The literature supplies the framework and universal bound, with no exact n=11 table located. Kalai's theory identifies these complexes as rational 2-hypertrees; later work treats their enumeration and torsion growth, while the exact eleven-vertex maximum remains unlocated.