# P12867: Effectively closed classes with computable maximal almost disjoint members

- ID: `P12867`
- Reference: `tianyuan-2026-effectively-closed-maximal-almost-disjoint`
- Page: https://theoremdb.org/statements/P12867
- 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/tianyuan-2026-effectively-closed-maximal-almost-disjoint?representation=markdown
- Record maturity: Reviewed problem

## The problem

Is there an effectively closed ($\Pi^0_1$) class $C\subseteq 2^{\mathbb N}$ such that its computable members form an infinite maximal almost disjoint family among the computable subsets of $\mathbb N$? Each member of that family must be infinite, distinct members have finite intersection, and every infinite computable subset of $\mathbb N$ must have infinite intersection with some computable member of $C$.

## 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 `tianyuan-2026-effectively-closed-maximal-almost-disjoint`, 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.