# P3650: A fourteen-vertex flag complex with second Betti number thirteen

- ID: `P3650`
- Reference: `flag-complex-fourteen-beta-two-thirteen`
- Page: https://theoremdb.org/statements/P3650
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/flag-complex-fourteen-beta-two-thirteen?representation=markdown
- Record maturity: Reviewed problem

## The problem

Does there exist a graph \(G\) on fourteen vertices with at most thirty-five edges whose clique complex satisfies \(\dim_{\mathbb F_2}H_2(\mathrm{Cl}(G);\mathbb F_2)\geq13\)?

### Remarks

- **Remark.** The clique complex Cl(G) has one simplex for every clique of G.
- **Remark.** Homology is computed over the field with two elements.

### What counts as a solution

- Give a graph with the required Betti number and an exact boundary-rank certificate, or exhaust all canonical fourteen-vertex graphs with at most thirty-five edges and certify that none qualifies.

## Status

**Resolution status:** Open.
**Lean verification:** Not Lean-verified.

A Lean formalization is attached. Signed verification is pending.

## Research packet

### Background and intake notes

The threshold is one above the best complete-multipartite construction under the edge budget, so either outcome improves a concrete extremal bound.

- The effect of edge addition on beta_2 is not monotone, so a simple supergraph pruning rule is unsound.
- Canonical graph generation is preferable to scanning labeled adjacency matrices. Store graph6 strings and boundary ranks for any near misses.

- Recorded example: The complete tripartite graph K_(3,3,4), together with four isolated vertices, has 33 edges and second Betti number 12.

### Computational notes

- The clique complex of K_(3,3,4) is the join of discrete sets of sizes 3, 3, and 4. The join homology formula gives beta_2=(3-1)(3-1)(4-1)=12 exactly.
- Complete multipartite constructions using fourteen vertices and at most thirty-five edges were enumerated over integer partitions; their largest beta_2 was 12.

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `flag-complex-fourteen-beta-two-thirteen`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.