Where the questions come from
The problem catalogue contains 8,052 source-listed entries across seven collections: 1,747 general research questions and 6,305 published parameter cases. A parameter case asks about a particular code, design, or Ramsey number within a larger mathematical family. These cases are not 6,305 independent general conjectures. The interface lets you browse either kind, and initially alternates collections so the largest family does not crowd out the others.
| Collection | Entries | Source observation |
|---|---|---|
| Erdős Problems | 590 | September 9, 2026 |
| Kourovka Notebook | 1,151 | September 1, 2026 edition |
| Millennium Prize Problems | 6 | September 10, 2026 |
| Linear Code Tables | 4,500 | September 10, 2026 |
| Constant-Weight Codes | 273 | September 10, 2026 |
| Covering Designs | 1,480 | July 23, 2026, via ConjectureBench |
| Ramsey Numbers | 52 | Tracker observed September 10, 2026 |
Each entry includes a source locator, topics, dated status, and an exploration-fit rating. No variants are generated to increase the count.
“Listed open” means that the selected source lists the problem as unresolved at the date shown. It is a source observation, not an independent literature review or a claim that the problem is still unsolved today. Consult the original source before starting research. Formal statements, formal proofs, and verified proofs are different things; this catalogue does not claim formal verification.
Erdős Problems · 590 entries
The Erdős Problems maintainer database is maintained by Thomas Bloom, Terence Tao, and contributors. The snapshot uses revision 3c68e941162f81d650fc886eed34e58bed3a6a01, dated September 9, 2026.
Only records whose informal_status.state is exactly open are eligible. Erdős #1163 is omitted because the maintainer explicitly flags its formulation as unclear; it remains linked as related context under #1162. Proved, disproved, solved, independent, and other status categories are excluded. Number, source status, tags, and any prize amount come from the maintainer feed. Prize terms and updates remain with the source.
Metadata is distributed under Apache License 2.0. The catalogue adds identifiers and short descriptions of the linked reference; it does not reproduce the website’s problem statements. A copy of the upstream license is included.
Kourovka Notebook · 1,151 entries
The Kourovka Notebook: Unsolved Problems in Group Theory, edited by E. I. Khukhro and V. D. Mazurov, is a research collection that has grown since 1965. Individual problem authors are credited in the notebook.
The imported canonical problem numbers come from ConjectureBench’s July 2026 index. Every retained identifier was checked against the editors’ September 2026 edition, announced in their September 1 update.
The import stops before the solved archive and excludes any problem containing an answer asterisk, including an answered subpart. This conservative rule omits some questions with unresolved parts. Each retained entry links to its page in the September edition. Kourovka 21.93 and Erdős #274 describe the same Herzog–Schönheim conjecture, so they are counted once, with both source links retained under Erdős #274. Topic labels are assigned from subject keywords to aid discovery; they are not claims about the mathematical content or difficulty.
The book remains copyright © E. I. Khukhro and V. D. Mazurov, 2026. Book text is not redistributed. Entries are bibliographic pointers with topic labels; read the original PDF for statements, assumptions, context, and author credits.
ConjectureBench attribution
The canonical Kourovka index was adapted from ConjectureBench, by Anirudha Ramesh and Shreyas Pimpalgaonkar, Bespoke Labs (2026), revision 357bcb1a1daf93917d42e8206ceaa55645729a09.
Its original curation metadata is licensed under CC BY 4.0. Ansatz retained canonical Kourovka identifiers, rechecked the September notebook, omitted book statements, excluded answers, and added topic labels. The expansion also retains numeric covering-design bounds from the pinned July snapshot, with that older observation date made explicit. This attribution does not relicense the underlying book. The upstream data license and source notice are preserved.
Millennium Prize Problems · 6 entries
The Clay Mathematics Institute’s list separates six unsolved problems from the solved Poincaré conjecture. Only those six unsolved entries are included, with links to the official formulations and the $1 million prize subject to CMI’s rules. Brief catalogue summaries are original navigation aids; official assumptions and prize rules remain authoritative.
Linear Code Tables · 4,500 parameter cases
Markus Grassl’s code tables record lower and upper bounds on the largest minimum distance of a linear code over a specified finite field. We parse every cell with two distinct numeric bounds in the downloaded ranges: GF(2), n ≤ 256 and k ≤ 14; GF(3), n ≤ 243 and k ≤ 12; GF(4), n ≤ 256 and k ≤ 9; GF(5), n ≤ 130 and k ≤ 8; GF(7), n ≤ 100 and k ≤ 7; GF(8) and GF(9), n ≤ 130 and k ≤ 6. These dimension ranges follow the published ConjectureBench source inventory; the source has larger tables too.
A single-number cell is excluded. Cells marked as missing their construction are also excluded. Links point to the exact q, n, k cell. Credit: M. Grassl, Bounds on the minimum distance of linear codes and quantum codes, together with A. E. Brouwer and the construction authors credited by the source. Only numerical bounds and parameters are included; generator matrices are not copied.
Constant-Weight Codes · 273 parameter cases
A. E. Brouwer’s tables give bounds for A(n,d,w). We include every explicitly unresolved lower–upper cell found in the downloaded tables for d ≥ 6, preserving the row n and column w. The d = 4 tables provide lower bounds alone and are excluded. Exact cells and the source’s red “lost construction” cells are excluded. Unlike the earlier index, this import has no codeword-count cap.
Credit: A. E. Brouwer; A. E. Brouwer, J. B. Shearer, N. J. A. Sloane and W. D. Smith, A new table of constant weight codes (1990); E. Agrell, A. Vardy and K. Zeger, Upper bounds for constant-weight codes (2000); and later contributors identified in the original table. Numeric bounds and coordinates are retained; actual codes are not redistributed.
Covering Designs · 1,480 historical parameter cases
The July 23, 2026 bounds come from ConjectureBench’s pinned La Jolla Covering Repository records. These entries are not rechecked current open problems. Their original website has retired, and the maintainer’s current page directs readers to the archived data and Giovanni Acerbi’s successor repository for later improvements. Each card retains the July date; each detail view explicitly says that current status was not rechecked.
A case specifies C(v,k,t), the minimum number of k-element blocks covering all t-element subsets. We include only recorded strict lower–upper gaps and preserve the source’s lower bound, which can be stronger than the elementary Schönheim bound. Credit goes to Daniel M. Gordon, the contributing construction authors, and ConjectureBench for its curation metadata. Only numeric bounds and bibliographic information are retained.
Ramsey Numbers · 52 parameter cases
The Leaps in Bounds tracker links its numeric bounds to the original researchers and, where applicable, Stanisław Radziszowski’s Small Ramsey Numbers survey. We retain its two-colour complete-graph Ramsey entries with integer lower and upper bounds that differ, excluding exact values. Each entry links to its own history and references.
The observation date is when the tracker was downloaded, not the publication date of a new mathematical result. Tracker histories can be incomplete. This import reports tracker bounds; it does not independently validate each proof or claim to have rechecked the whole survey. Only numeric values and parameters are reproduced.
Exploration-fit ratings
Every entry receives an AI-assisted editorial heuristic from 1 to 5, measuring how concrete a starting point the available metadata provides for agent-assisted research. Higher scores suggest more explicit experimental targets. They do not measure mathematical importance, calibrated difficulty, proof quality, probability of success, or expected time to solve. A famous general conjecture and one finite case ask different kinds of questions.
| Score | Label | Basis |
|---|---|---|
| 1/5 | Long-term theory | A broad Millennium problem; first narrow a special case. |
| 2/5 | Literature first | A bibliographic research question whose statement and context need inspection. |
| 3/5 | Specialist exploration | An explicit finite target beyond the verification-size threshold below. |
| 4/5 | Concrete experiments | An explicit finite target within the verification-size threshold. |
| 5/5 | Focused experiments | A level-4 target with recorded integer bounds one apart. |
The finite-case thresholds are deliberately simple: at most 100,000 codewords for a linear code; at most 1,000,000 pairwise distance checks for a one-step constant-weight construction; at most 100,000 t-subsets for a covering; and a recorded Ramsey lower bound at most 60 vertices. These are estimates of basic verification scale, not construction or search cost. Exploiting algebraic structure may change the practical cost dramatically. A gap of one can still be exceptionally hard.
For example, the binary length-65, dimension-9 code case has 512 codewords and recorded distance bounds 28–29, so it receives 5/5. A one-unit gap over an enormous space receives 3/5. Uninspected Erdős and Kourovka pointers receive 2/5 with limited metadata; this does not assert that they are easy, hard, or unimportant. Click any card to see its reason and confidence. The deterministic rubric and version are embedded in the dataset, so the ratings can be reproduced and revised.
Reproducibility and updates
The public and workspace catalogues use identical schema-v2 JSON. All current entries live in problems; compact_families is empty. Use all_records in scripts/problem_expansion.py to stream full records. The public page downloads /research/open-problems.json over the existing compressed route and builds a search index; the historical /open-problems.json URL remains available.
The snapshot records revisions, input SHA-256 hashes, attribution, and exact counts. The importers are scripts/import_open_problems.py and scripts/problem_expansion.py; they read local data and never execute upstream scripts. Expansion inputs and their checksums are preserved in artifacts/open-problems-expansion-2026-09-10/. Collection-removal checks are recorded in artifacts/remove-difference-sets-2026-09-10/; the importer follows the current seven-collection selection.
To reproduce this snapshot, install PyYAML and pypdf, obtain the pinned Erdős data/problems.yaml, extract the pinned ConjectureBench archive, and download the September notebook PDF. Then run:
python3 scripts/import_open_problems.py \
--erdos /path/to/problems.yaml \
--catalog /path/to/conjecture-bench \
--notebook /path/to/21tkt.pdf \
--expansion-sources artifacts/open-problems-expansion-2026-09-10/sourcesAn update requires reviewing source revisions and status dates in the importer, regenerating both snapshots, and running tests/test_open_problem_catalogue.py, tests/test_problem_expansion.py, and node --test tests/test_problem_catalogue.mjs. The interface does not present a historical snapshot as a live status feed.
Community contributions
Anyone can read the forum. Signed-in users can propose a problem, start a discussion, or reply. Proposed problems are stored as unreviewed community submissions and do not automatically become source-verified catalogue entries or harness memory. Please include the original statement, a source link where available, and what you believe remains open. Discussions can also report a source correction or a newly published result.