# Erdős Problem 1007

A statement in qed.bot, the register of claims of AI work in mathematics. The register grades the evidence attached to each claim; it never rules on whether a proof is correct.

- Identifier: erdos:1007
- Collection: Erdős
- Page: https://qed.bot/s/erdos-1007
- Formal record: solved; at its source: solved (Lean)
- Machine-checked: yes, by Palomar
- Fidelity F3: The project checks its proof with Comparator against a statement from a corpus written separately from the proof, and held here.
- Source: https://www.erdosproblems.com/1007

## AI claims

### Aristotle, 19 Jan 2026

New proof found (Lean)

- Outcome: full
- Autonomy A2: Directed. The system produced the result while building on literature or framing supplied to it.
- Evidence V3: Machine-checked. The artifact was rebuilt at a pinned commit by someone other than its authors — qed.bot, or a registry that publishes the check — and it holds: a proof that elaborates on the standard axioms alone, or an object that meets the problem's own constraints. Checked by Palomar.
- Fidelity F3: Anchored. The proof is checked for exact statement identity against a statement written independently of it, held here from a separate statement corpus.
- Source: Erdős problems AI wiki, https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems

### Aristotle, 19 Jan 2026

House (2013)

- Outcome: full (a supporting task)
- Autonomy A1: Collaborative. A human and a system worked the problem together, or the system played a supporting role.
- Evidence V3: Machine-checked. The artifact was rebuilt at a pinned commit by someone other than its authors — qed.bot, or a registry that publishes the check — and it holds: a proof that elaborates on the standard axioms alone, or an object that meets the problem's own constraints. Checked by Palomar.
- Fidelity F3: Anchored. The proof is checked for exact statement identity against a statement written independently of it, held here from a separate statement corpus.
- Source: Erdős problems AI wiki, https://github.com/teorth/erdosproblems/wiki/AI-contributions-to-Erd%C5%91s-problems

## Checks

- verified, by Palomar: Registered by Palomar at 43f89415: Comparator confirmed 1 theorems prove the recorded statement within Palomar's axiom policy, replayed through Lean's kernel and the independent nanoda kernel (https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-23-000003)
- verified, by Palomar: Registered by Palomar at 85032a43: Comparator confirmed 1 theorems prove the recorded statement within Palomar's axiom policy, replayed through Lean's kernel and the independent nanoda kernel (https://palomar-registry.org/entry.html?id=PALOMAR-2026-09-25-000003)

## Formal statements

- FormalConjectures/ErdosProblems/1007.lean, https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/1007.lean

Cite as: qed.bot, "Erdős Problem 1007", https://qed.bot/s/erdos-1007, as of 30 Sep 2026. The register's data is published under CC BY 4.0: https://creativecommons.org/licenses/by/4.0/