# P11526: Which Johnson graphs have Catalan clique-covering number?

- ID: `P11526`
- Reference: `catalan-clique-covers-of-johnson-graphs`
- Page: https://theoremdb.org/statements/P11526
- 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/catalan-clique-covers-of-johnson-graphs?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(J(2k,k)\) be the Johnson graph whose vertices are the k-subsets of a 2k-element set, with two vertices adjacent when their intersection has size k-1. Let theta(G) be the minimum number of cliques whose vertex sets cover V(G). For which positive integers k does theta(J(2k,k)) equal the k-th Catalan number C_k? In particular, does equality hold for every even k?

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `catalan-clique-covers-of-johnson-graphs`, 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.