Erdős Problems Database

• Erdős Problems Database is a digital repository that archives and tracks hundreds of mathematical problems posed by Paul Erdős.
• It integrates formal problem statements, bibliographic metadata, and status flags with AI-assisted and expert-reviewed workflows.
• The platform benchmarks automated discovery methods and supports research in combinatorics, geometry, number theory, and Ramsey theory.

The Erdős Problems Database is a curated, structured digital repository cataloguing hundreds of open, solved, and partially resolved mathematical problems originally posed or popularized by Paul Erdős. Its primary function is to serve as a reference point for combinatorics, discrete geometry, analytic number theory, and Ramsey theory, as well as to benchmark advances in both human and automated mathematical reasoning. The database integrates problem statements, literature references, status tracking, and, increasingly, metadata supporting semi-autonomous triage and discovery workflows.

1. Database Origin, Scope, and Philosophy

Launched in 2023 by Thomas Bloom, the canonical Erdős Problems Database (erdosproblems.com) systematically records 1,179 distinct problems attributed to Erdős, based on his published papers, problem books, conference pronouncements, and scholarly correspondence. Each database entry includes a verbatim problem statement, bibliographic links, progress notes, and a status flag—usually “Open” or “Solved” (41% solved, 59% open as of early 2026) (Feng et al., 29 Jan 2026). Coverage explicitly spans problems with strong contemporary relevance to combinatorics, geometry, number theory, and graph theory, as well as intellectually influential but technically unresolved conjectures.

The “Open” status denotes that no solution was known to at least one professional after checking standard resources; this flag is not a mathematical assertion but reflects current literature awareness. Obscurity of the literature, lack of cross-referencing, or erroneous transcription often explain why some entries remain marked “Open” after solutions have appeared elsewhere (Feng et al., 29 Jan 2026).

2. Structural and Methodological Features

2.1 Data Architecture

Each problem entry comprises:

• The problem in a formal statement (often LaTeX-encoded)
• Context and bibliographic metadata
• A status flag with optional elaboration (“Open,” “Solved,” “Partial,” “Conditional”)
• References, including primary and secondary literature
• Where available, links to complete solutions, partial advances, or significant counterexamples

The database format is explicitly designed for integration with programmatic verification and search tools (JSON, LaTeX markup) and is intended to be future-proof against anticipated advances in mathematical language processing.

2.2 Review and Update Protocols

Problem status is periodically re-evaluated. This may occur either after manual identification of a relevant publication, submission from the mathematical community, or via semi-automated literature mining facilitated by tools such as Gemini (Aletheia), which combine AI-driven candidate discovery with human expert validation (Feng et al., 29 Jan 2026).

AI-based curation follows a hybrid two-stage pipeline:

• Stage A: Automated natural language solution generation, plausibility scoring, and high-confidence filtering using a specialized verifier (NLV).
• Stage B: Manual assessment by non-specialists (rapid triage), followed by expert domain audit for correctness, novelty, and fidelity to original intent (Feng et al., 29 Jan 2026).

A meaningful resolution (solution, counterexample, sharp partial result) is only incorporated after complete expert audit, which may result in status change, database annotation, or integration of additional references.

3. Use in Mathematics Research and Discovery

The database is both a resource and a testbed:

• Primary Reference: For many open conjectures, especially those informally transmitted, the Erdős Problems Database functions as the de facto canonical source of the “current” statement and known progress (Feng et al., 29 Jan 2026).
• Benchmark for Automated Discovery: The database has become a standard benchmark suite for semi-automated (AI-assisted) solution pipelines. In a case study involving 700 “Open” problems, a hybrid AI+human methodology yielded 13 correct or rediscovered solutions, with a notable fraction attributable to prior literature that was only obscure rather than unsolved (Feng et al., 29 Jan 2026).
• Curatorial Platform: The curation workflow incorporates AI-accelerated triage with rigorous human oversight. This hybrid approach revealed that a nontrivial percentage (estimated 5–10%) of “Open” entries may actually have already-resolved status due to overlooked or badly cross-referenced literature.
• Integration with Bibliographic Services: There is ongoing migration toward integration with MathSciNet, zbMATH, and citation extraction algorithms for automated status-check and bibliographic updates (Feng et al., 29 Jan 2026).

4. Challenges and Ongoing Developments

4.1 Literature Identification and Statement Fidelity

Identifying prior solutions or closely related results is nontrivial, particularly for questions not directly indexed in standard abstracts or for those solved in non-traditional outlets (blogs, preprints, national olympiad materials, etc.).

Risks include “subconscious plagiarism” by AI (solutions that closely match previously published work without attribution) and status misclassification due to subtle differences in the precise Erdős statement versus similar-looking, solved variants (Feng et al., 29 Jan 2026). Best practices now demand strict logging of solution traces, verifiable provenance, and independent human audit.

4.2 Maintenance and Workflow Recommendations

The ongoing update strategy is predicated on:

• Tiered AI-verification workflows: rapid LLM-based screening, human-in-the-loop judgment, and comprehensive expert audit.
• Explicit linking to standardized bibliographic APIs for dynamic cross-referencing.
• A community-driven “claim tracking” system for versioning discoveries, solutions, and attempted or claimed advances in real time.
• Integration of formal-verification and proof-assistant (e.g. Lean) capabilities for new candidate solutions.

5. Impact, Limitations, and Future Directions

The Erdős Problems Database now serves as a nexus for both the human and automated mathematical community. It is a proving ground for methodologies leveraging hybrid reasoning, literature mining, and AI-human interaction in conjecture triage.

While current impact is limited by coverage, literature obscurity, and the challenge of precise statement-matching, the rate of solution and clarification for previously open problems is increasing as verification pipelines mature. The process has illuminated the importance of expert intervention in problem fidelity and status updating.

Future computational directions include automated proof-obligation generation, formal statement parsing for large-scale status auditing, and collective “claim wikis” to close the gap between informal folklore, preprints, and canonical open-problem registries. Ongoing curation is critical for ensuring that open-problem status reflects actual mathematical uncertainty rather than bibliographic obscurity (Feng et al., 29 Jan 2026).

6. Selected Workflow Schematic

for problem in ErdosProblems.Open: sol = Gemini.generate_solution(problem.prompt) score = NLV.score(sol) if score > τ: candidates.append((problem,sol)) human_quick_filter = filter(candidates, by_quick_human_judgment) for (prob,sol) in human_quick_filter: domain_result = domain_expert_audit(prob,sol) if domain_result.correct: record(domain_result.category, prob, sol)

Editor's term: “Hybrid AI curation pipeline”—the sequential application of automated reasoning, plausibility filtering, and human domain audit in database maintenance (Feng et al., 29 Jan 2026).

• "Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems" (Feng et al., 29 Jan 2026)—methodologies, database structure, workflow, and meta-summary of curatorial pipelines.
• www.erdosproblems.com—the canonical online database implementation, with real-time status updates, problem indexing, and links to primary literature.

---

In summary, the Erdős Problems Database is an expanding, cross-referenced, expert-curated computational platform for the archival, status tracking, and systematic triage of hundreds of significant open and solved problems in contemporary mathematics. Its structure and maintenance protocols are at the frontier of hybrid human–machine mathematical discovery and open-problem management.