graph-conjectures

Resolved-conjecture timeline: the illustration predates the Bondy–Murty corpus and later literature-status corrections; the site's /timeline/ page currently shows 130 dated solved or disproved entries. Editable SVG: scraper/static/resolved-conjectures-timeline.svg.

A browseable, status-annotated mirror of the graph-theory category of openproblemgarden.org, extended with new conjectures mined from recent arXiv papers by 12 curated graph-theorists. Every problem and conjecture is classified as open, partial, solved, disproved, or unclear via an automated literature- review pass, with every cited paper verified by WebFetch before inclusion — and connected by a relation graph of AI-reviewed implications, equivalences, and duplicates, browsable interactively on the site (details).

The site also imports literature-status corrections from already_resolved results and full-problem ALREADY_KNOWN referee verdicts in Graph-Theory-LLM-Proofs. They appear as solved or disproved according to the result, with a reference to the existing article credited for the resolution and a separate link to the automated audit. No model-generated proof is promoted to solved. Refresh the data from sibling checkouts with python scripts/sync_llm_proof_results.py.

The same import exposes only PDF write-ups that are listed in that repository's README and received a CONFIRMED referee verdict. Results marked MINOR_GAPS are excluded. These model-generated results use the separate ai-proved or ai-disproved labels and link the corresponding PDF. If the literature already settles the problem, the ordinary literature status takes precedence while the AI artifact remains available on the detail page.

The same two labels record claims from OpenAI's October 2026 collection of model-generated manuscripts (github.com/openai/math, 722 manuscripts in 372 result families). Matches against the catalogue are made by hand, after reading each manuscript's TeX source, and listed in scripts/sync_openai_math_results.py. An entry moves to ai-proved or ai-disproved only when a manuscript settles its exact statement, either directly or through a short argument recorded on the page. Partial progress is linked without changing the status. See OpenAI's manuscript collection.

The site also displays a separate ill-posed diagnostic on 23 arXiv records flagged by the automated attacks in Graph-Theory-LLM-Proofs. These unrefereed model self-reports do not overwrite the literature-review status; each affected detail page links to the complete attack artifact and includes its reason and caveat.

Four corpora, one merged index:

corpus	count	scope
OPG	227	full graph-theory tag of openproblemgarden.org
arXiv	768	new conjectures from 857 arxiv papers (2016–2026) by 12 curated authors
Bondy–Murty	38	items of Appendix A (100 "unsolved problems") of Bondy & Murty's Graph Theory that the two corpora above did not cover (cross-check)
others	1	conjectures studied in a problems/ workstream but absent from OPG and the arXiv extraction (currently Meyniel's cop-number conjecture)

Reviewed status counts:

status	OPG	arXiv	Bondy–Murty	meaning
open	51	514	12	no significant progress found in literature search
partial	127	118	21	progress since posting
ai-proved	6	10	0	model-generated proof (confirmed write-up or OpenAI manuscript)
ai-disproved	6	12	1	model-generated counterexample/disproof (same sources)
solved	23	83	3	fully proved in the literature
disproved	13	22	1	counterexample in the literature
unclear	1	3	0	insufficient information to decide

One OPG problem (OPG_tournament_feedback_arc_set_ptas, ptas_for_feedback_arc_set_in_tournaments) counts as solved although it was already solved before it was posted on OPG (Kenyon-Mathieu–Schudy, STOC 2007). Its review has solved_before_posting: true, and its page says so.

The most recent status changes come from OpenAI's manuscript collection (October 2026). Thirteen entries moved to ai-proved or ai-disproved, among them Hadwiger's conjecture (counterexample), Barnette's and Seymour's second-neighbourhood conjectures, Sidorenko's and Ryser's conjectures (counterexamples), and the Harary–Hill and Zarankiewicz crossing-number formulas. See Recently claimed. Before that, the cycle double cover conjecture (OPG_bridgeless_cycle_double_cover, /op/cycle_double_cover_conjecture/) moved partial → solved in September 2026 after the July 2026 OpenAI proof and Sang-il Oum's exposition (arXiv:2607.16356); see Recently resolved.

44 of the arXiv-extracted records were also identified as direct progress on existing OPG problems and attached as citations on those OPG pages (e.g. on OPG_caccetta_haggkvist_outdegree_short_cycle (/op/caccetta_haggkvist_conjecture/) and 20 other pages).

The output site is inspired by erdosproblems.com, which itself is an excellent companion source.

Canonical conjecture names

Every record of the four corpora carries a canonical short name of the form PREFIX_english_name, where PREFIX is OPG, AX, BM or OTH and english_name is a few lowercase words conveying the statement (e.g. OPG_bridgeless_cycle_double_cover, AX_cop_number_sublinear_treedepth, BM_hypohamiltonian_min_degree_4, OTH_meyniel_cop_number_sqrt_n). The 1034 names are distinct; slight variants of one statement share a leading stem and differ by a trailing qualifier. The old identifiers (OPG slug, arXiv review id, bm_id, others id) are unchanged and remain the keys for data files and sibling repositories. Site pages are addressed by the canonical name (/bm/BM_cubic_second_hamilton_cycle_polynomial/); the old address (/bm/bm-021/) redirects there; the result files data/llm_proof_results.json and data/openai_math_results.json are keyed by them and also carry the name. The correspondence old id → new name is in CONJECTURE_NAMES.md and, machine-readable, in data/conjecture_names.json.

Relations between conjectures

The corpus also carries a verified relation graph, data/relations.json: 225 adversarially verified relations between conjectures across the corpora — 159 implications ("if A is true then B is true"), 8 equivalences, 16 cross-corpus duplicates, and 42 documented non-implication links. Each edge records its verified direction, a checkable referee argument, and citations where the confirmation is literature-based. It was produced by a multi-agent find-and-refute pipeline (17 finder agents, then 59 refute-by-default verifiers; 17 of 207 candidates were refuted, including two false fuzzy matches and a conjecture-name collision); 190 edges survived, and 4 duplicate pairs the phase-B dedup had missed were merged in September 2026. An incremental pass then connected the Bondy–Murty and others corpora, adding 39 edges (34 confirmed, 5 plausible) out of 42 candidates, plus one follow-up edge (orientable 5-CDC ⇒ 5-flow). "Verified" means checked by adversarial AI referees, not formally verified — no argument has been machine-checked in a proof assistant such as Rocq or Lean. Confronting each edge with the review statuses of its two endpoints is now done at build time and shown on the site: 42 implications are vacuous (target already a theorem, or source already disproved — 10 of them because the cycle double cover conjecture was proved), 4 implications are to verify (a solved source or a disproved target would settle the other endpoint, but the implication is only AI-checked, so it must be formally verified or peer reviewed before that status can change), and 1 duplicate edge is inconsistent (its endpoints have different statuses). See RELATIONS.md for the design, findings, and caveats; pipeline provenance lives in data/relations_work/.

Recently claimed: OpenAI's manuscript collection (October 2026)

On 6 October 2026 OpenAI released github.com/openai/math: 722 manuscripts in 372 result families, produced by an unreleased internal model and not peer reviewed. OpenAI's Lean scope pages (lean/docs/<family>.md) name the manuscripts whose main result is formalized. The formalization catalogue as a whole is marked review: unchecked, and this project has not run its Comparator checks.

On 7 October 2026 every family in Combinatorics and Theoretical computer science, plus the graph-related families from other sections, was compared with all 1034 catalogue entries (commit adc7f12). Each match was checked against the manuscript's TeX source. Status changes:

page	before	after	OpenAI manuscript	Lean
BM_hadwiger_minor_chromatic (bm-041) Hadwiger's conjecture	open	ai-disproved	A counterexample to Hadwiger's conjecture: $\alpha(G)\le2$ and $h(G)<26m/75+2/3<m/2\le\chi(G)$	no
OPG_seagull_independence_2_clique_minor_half_n (seagull_problem)	partial	ai-disproved	same graphs: no clique minor on $m/2$ vertices	no
AX_kt_minor_free_independence_n_over_t (1907.12999__00) ($\alpha\ge n/t$ without a $K_{t+1}$ minor)	open	ai-disproved	same graphs, with $t=h(G)$	no
OPG_hadwiger_list_kt_minor_free_choosable (list_hadwiger_conjecture)	partial	ai-proved	A linear list-coloring bound in terms of the Hadwiger number	yes
OPG_sidorenko_bipartite_homomorphism_density (sidorenkos_conjecture)	partial	ai-disproved	A counterexample to Sidorenko's conjecture (35 vertices, 66 edges)	yes
AX_directed_sidorenko_bipartite_homomorphism (2210.16971__00), AX_directed_sidorenko_forcing_cycle_homomorphism (2210.16971__01) (directed Sidorenko and forcing)	open	ai-disproved	same graph, through Theorems 1.5 and 1.8 of the source paper	Sidorenko part only
OPG_ryser_r_partite_hypergraph_cover_matching (rysers_conjecture)	partial	ai-disproved	two papers: intersecting $r$-partite hypergraphs with $\tau=r$	yes
AX_r_partite_hypergraph_matching_drop_deletion (2505.05339__02) (deletion form implying Ryser)	open	ai-disproved	same hypergraphs	yes
OPG_crossing_number_complete_graph_harary_hill (the_crossing_number_of_the_complete_graph)	partial	ai-proved	The crossing number of complete graphs (Harary–Hill)	yes
OPG_crossing_number_complete_bipartite_zarankiewicz (the_crossing_number_of_the_complete_bipartite_graph)	partial	ai-proved	The crossing number of complete bipartite graphs (Zarankiewicz)	yes
OPG_seymour_second_neighborhood_outdegree (seymours_second_neighbourhood_conjecture)	partial	ai-proved	A proof of Seymour's second-neighborhood conjecture	yes
OPG_barnette_cubic_planar_bipartite_hamiltonian (barnettes_conjecture)	partial	ai-proved	Paired states and Hamiltonian cycles in cubic bipartite planar graphs	yes

Nine further entries link a manuscript as partial progress, with their status unchanged:

• BM_hadwiger_nelson_plane_unit_distance_chromatic (bm-054): the chromatic number of the plane is 6 or 7.
• OPG_caccetta_haggkvist_outdegree_short_cycle (caccetta_haggkvist_conjecture): the second-neighbourhood proof gives a cycle of length at most 3 when both minimum degrees are at least $n/3$, and the girth-four case of Behzad–Chartrand–Wall. Both are listed as open on the page. The case $r=n/3$, which bounds only the outdegree, stays open.
• OPG_hadwiger_fractional_chromatic_minor (fractional_hadwiger): part (a) fails; parts (b) and (c) are not addressed.
• AX_kt_minor_free_list_chromatic_2t (2110.09403__00), AX_kst_minor_free_choosable_2s_t (2201.09115__01): linear list-colouring bounds, but the constants $2t$ and $2s+t$ remain open.
• AX_kt_minor_free_hypergraph_chromatic_bound (2206.13635__00): $h(t)=O(t)$ for hypergraphs.
• OPG_eulerian_cycle_decomposition_hajos_n_half (decomposing_an_eulerian_graph_into_cycles): $O(n)$ cycles; Hajós's bound remains open.
• AX_crossing_number_kn_minus_matching (2009.03418__00): only the $t=0$ case.
• AX_ordered_removal_lemma_binary_matrices_polynomial (1704.02367__01): polynomial removal is ruled out; exponential dependence is open.

Some entries are not settled despite overlapping topics. BM_gyarfas_triangle_free_infinite_chromatic_induced_trees (bm-050) (Gyárfás–Sumner for triangle-free graphs) is a different statement from the Gyárfás tree-cover conjecture that family 162 disproves. BM_hajos_k5_k6_subdivision_chromatic (bm-042) (Hajós for $k=5,6$) is untouched.

The first three changes rest on the Hadwiger counterexample. That manuscript is outside OpenAI's Lean formalization of family 157, which covers only the list-colouring bound. The forcing corollary used for AX_directed_sidorenko_forcing_cycle_homomorphism (2210.16971__01) is likewise outside the Lean scope of family 161. The ai-* labels keep these claims apart from the literature so they can be revisited once expositions or referee reports appear, as happened for the cycle double cover conjecture below.

Recently resolved: the cycle double cover conjecture

On 11 July 2026 OpenAI announced a proof, produced by GPT-5.6 Sol Ultra, that every bridgeless graph has a cycle double cover, settling the conjecture of Seymour and Szekeres. Sang-il Oum's exposition, arXiv:2607.16356, presents it in a self-contained ten pages assuming only Fleischner's splitting lemma and the Nash-Williams–Tutte tree-packing theorem: reduce to cubic 3-edge-connected graphs, take a nowhere-zero $\mathbb{Z}_2^3$-flow $\phi$, and use an $\mathbb{F}_2$ consistency criterion to choose vertex labels $t_v$ making the two-element edge label $P_e = t_u + \phi(f_u) + \langle\phi(e)\rangle$ well defined; the supports of the seven nonzero labels are Eulerian subgraphs covering every edge exactly twice.

What this repository changed as a result (September 2026):

page	before	after
OPG_bridgeless_cycle_double_cover (cycle_double_cover_conjecture)	partial	solved
OPG_2_connected_circular_surface_embedding (the_circular_embedding_conjecture)	partial	partial — but the cubic case is now settled
OPG_bridgeless_cycle_double_cover_5_cycles (m_n_cycle_covers)	partial	partial — the (8,2) box of the chart is now a theorem
OPG_bridgeless_cycle_double_cover_strong_5 (strong_5_cycle_double_cover_conjecture), OPG_faithful_cycle_cover_even_weights (faithful_cycle_covers), OPG_bridgeless_cycle_double_cover_prescribed_2_factor (cycle_double_covers_containing_predefined_2_regular_subgraphs), OPG_eulerian_cycle_decomposition_compatible_transition_system (decomposing_eulerian_graphs), OPG_petersen_coloring_bridgeless_cubic (petersen_coloring_conjecture)	partial/open	unchanged; their implication into CDC is now flagged vacuous
OPG_berge_fulkerson_cubic_6_perfect_matchings (the_berge_fulkerson_conjecture)	partial	unchanged; the related_only reading is confirmed — CDC transfers nothing to it

Explicitly not settled by the proof, and still open on the site: the 5-cycle double cover conjecture (Celmins, Preissmann — the (5,2) case of OPG_bridgeless_cycle_double_cover_5_cycles (m_n_cycle_covers)), the orientable 5-CDC conjecture (Archdeacon, Jaeger), which would imply the 5-flow conjecture, the oriented cycle double cover conjecture, Bondy's $n-1$ small cycle double cover conjecture for simple 2-edge-connected graphs, the strong embedding conjecture for non-cubic graphs, and Berge–Fulkerson. Bondy's cubic small CDC conjecture does become unconditional, via Lai–Yu–Zhang, and the Jamshy–Tarsi matroid equivalence upgrades to a theorem about coloop-free binary matroids with no $F_7^*$-minor.

The proof has not been peer-reviewed or journal-published, and CDC has a history of flawed arXiv proofs; the solved status is recorded with that caveat spelled out in the reviewer notes on the page.

Quick start

# 1) install deps (uv recommended)
uv venv
uv pip install -r scraper/requirements.txt

# 2) the data is already in data/; render the static site
.venv/bin/python scraper/build.py --confirmed-only
python -m http.server --directory site 8000
# open http://localhost:8000

Curated arXiv authors

data/arxiv_authors.json lists the 12 authors whose recent (since 2016) work on math.CO, cs.DM, and math.GT is mined for new conjectures:

Paul Seymour · Sophie Spirkl · Stéphan Thomassé · Pierre Aboulker
Alex Scott · Zdeněk Dvořák · Jacob Fox · Noga Alon
Gwenaël Joret · Raphael Mario Steiner · Bojan Mohar · Marthe Bonamy

Edit that file to grow or shrink the curated set; the rest of the pipeline re-runs idempotently.

Pipelines

A. OPG mirror (227 problems)

.venv/bin/python scraper/crawl.py          # polite OPG crawler, 12s/req
.venv/bin/python scraper/parse.py          # HTML → data/problems.json
.venv/bin/python scraper/erdos_index.py    # erdosproblems.com index
.venv/bin/python scraper/intersect.py      # OPG ↔ erdosproblems cross-ref
# Literature review (8-terminal worker pattern):
.venv/bin/python scripts/partition.py --workers 8
# In 8 terminals: PER_SLUG_TIMEOUT=900 WORKER=NN bash scripts/run_worker.sh

B. arXiv mining (768 new conjectures)

# (1) Metadata harvest via OAI-PMH — fast, polite, ~7 min for the 3 sets
.venv/bin/python scripts/arxiv_oai_fetch.py --set math.CO,cs.DM,math.GT --from 2016-01-01

# (2) Download paper content (HTML, PDF fallback) — ~2h for 857 unique papers
.venv/bin/python scripts/arxiv_download_content.py

# (3) Partition + 8-worker extraction via `claude -p` — ~2h on Max plan
.venv/bin/python scripts/arxiv_partition.py --workers 8
# In 8 terminals: PER_PAPER_TIMEOUT=900 WORKER=NN bash scripts/arxiv_run_worker.sh

# (4) Aggregate: dedup states + fuzzy-match studies → OPG
.venv/bin/python scripts/arxiv_aggregate.py
# (review data/arxiv_opg_matches.tsv, set manual_confirmed flags as desired)

# (5) Phase 1 internal cross-reference (free, no claude calls, ~1s)
.venv/bin/python scripts/arxiv_internal_refs.py

# (6) Phase 2 web-search status review — ~5h on Max plan
.venv/bin/python scripts/arxiv_review_partition.py --workers 8
# In 8 terminals: PER_REVIEW_TIMEOUT=900 WORKER=NN bash scripts/arxiv_review_run_worker.sh

C. Bondy–Murty Appendix A (38 conjectures)

Appendix A of Bondy & Murty's Graph Theory (GTM 244, 2008; French edition by F. Havet, 2025, HAL) lists 100 open problems. BONDY_MURTY_APPENDIX_A.md cross-checks every item against the OPG and arXiv corpora: 60 were already present, 6 only weakly covered, 32 missing, 2 out of scope. The 38 uncovered items form the third corpus.

# (1) Regenerate the records (hand-transcribed statements live in the script)
.venv/bin/python scripts/bm_build_records.py       # → data/bondy_murty_conjectures.json

# (2) Status review via `claude -p` with web search, same schema as the arXiv reviews
PER_REVIEW_TIMEOUT=900 .venv/bin/python scraper/bm_review.py --all --jobs 6
#                                                   # → data/bondy_murty_reviews/<bm_id>.json

Each record keeps the appendix number, the book section and page, the English statement, the attribution, and related pointers to the nearest OPG / arXiv / erdosproblems.com records. Pages render under /bm/<bm_id>/.

Other workstream conjectures. data/others_conjectures.json holds hand-written records with the same shape (plus a workstream path) for conjectures that a problems/ workstream studies but that neither OPG nor the arXiv extraction contains. The same driver reviews them:

.venv/bin/python scraper/bm_review.py --records data/others_conjectures.json \
    --out-dir data/others_reviews --system-prompt scraper/others_review_system_prompt.md --all

Pages render under /others/<id>/ with a link back to the workstream.

D. OpenAI manuscript matches

The hand-checked matches live in MATCHES in scripts/sync_openai_math_results.py. The script resolves each listed manuscript against a checkout of the OpenAI repository, adding the PDF link pinned to the checked commit and the Lean flag from lean/docs/<family>.md:

git clone https://github.com/openai/math ../openai-math
.venv/bin/python scripts/sync_openai_math_results.py --source-dir ../openai-math
#                                                   # → data/openai_math_results.json

E. Site build

.venv/bin/python scraper/build.py --confirmed-only

--confirmed-only attaches only arxiv_opg_matches.json entries flagged manual_confirmed: true. Without the flag, auto-high (score ≥ 92) matches are also attached.

Architecture

OPG branch:
  crawl.py        →  cache/op/<slug>.html     polite crawl, 12s/req
  parse.py        →  data/problems.json       227 OPG problems
  erdos_index.py  →  data/erdos_graph.json    277 erdosproblems records
  intersect.py    →  data/intersection.json   fuzzy OPG ↔ erdos matches (only `confirmed: true` rows used)
  review run      →  data/reviews/<slug>.json 227 status reviews

arXiv branch:
  arxiv_oai_fetch.py      → cache/arxiv/<slug>/papers.json   12 author manifests
  arxiv_download_content  → cache/arxiv/<slug>/*.html|*.pdf  857 unique papers
  arxiv_run_worker.sh     → data/arxiv_extracted/*.json      1,565 raw records
  arxiv_aggregate.py      → data/arxiv_conjectures.json      768 deduped conjectures
                          → data/arxiv_opg_matches.{json,tsv} 44 confirmed citations
  arxiv_internal_refs.py  → data/arxiv_internal_refs.json    148 cross-corpus refs
  arxiv_review_*          → data/arxiv_reviews/<id>.json     762 status reviews

Bondy–Murty branch:
  bm_build_records.py     → data/bondy_murty_conjectures.json 38 appendix items not covered above
  bm_review.py            → data/bondy_murty_reviews/<id>.json 38 status reviews

Others branch:
  data/others_conjectures.json                              hand-written workstream conjectures
  bm_review.py --records … → data/others_reviews/<id>.json  status reviews

AI results:
  sync_llm_proof_results.py    → data/llm_proof_results.json   Graph-Theory-LLM-Proofs import
  sync_openai_math_results.py  → data/openai_math_results.json OpenAI manuscript matches

Site:
  build.py        →  site/                    Jinja2 → static HTML, KaTeX
                     /                        merged index (995 rows, filter by source/status)
                     /op/<slug>/              227 OPG problem pages
                     /arxiv/<id>/             768 arxiv conjecture pages
                     /bm/<bm_id>/             38 Bondy–Murty appendix pages
                     /others/<id>/            other workstream conjecture pages
                     /author/<slug>/          218 author landing pages
                     /tag/<slug>/             227 subject pages
                     /relations/              interactive relation-graph drawing
                                              (from data/relations.json via
                                              scraper/relations_layout.py; problem
                                              pages get a "Related conjectures" box)

Both review runs use a terminal-per-bucket worker pattern (scripts/*_run_worker.sh) rather than in-process parallelism. Each terminal runs claude -p sequentially for its bucket, with an os.killpg-based hard timeout per item to recover from infrastructure stalls. See LIT_REVIEW.md for the design rationale.

Repository layout

graph-conjectures/
├── scraper/                          # Python pipeline + Jinja2 templates + CSS
│   ├── crawl.py / parse.py           # OPG crawler + parser
│   ├── erdos_index.py / intersect.py # OPG ↔ erdosproblems cross-reference
│   ├── arxiv_fetch.py                # legacy /api/query author search (kept for reference)
│   ├── arxiv_extract.py              # claude -p extraction of one paper
│   ├── arxiv_review.py               # claude -p status review of one conjecture
│   ├── bm_review.py                  # claude -p status review of Bondy–Murty items
│   ├── arxiv_system_prompt.md        # extraction prompt
│   ├── arxiv_review_system_prompt.md # review prompt
│   ├── bm_review_system_prompt.md    # Bondy–Murty review prompt
│   ├── others_review_system_prompt.md # "others"-record review prompt
│   ├── review_system_prompt.md       # OPG-review prompt
│   ├── build.py                      # site generator
│   ├── templates/                    # Jinja2 templates
│   ├── static/style.css
│   └── requirements.txt
├── scripts/                          # worker-pattern harness
│   ├── partition.py / run_worker.sh  # OPG-review workers
│   ├── arxiv_oai_fetch.py            # OAI-PMH metadata harvest
│   ├── arxiv_download_content.py     # polite HTML/PDF download
│   ├── arxiv_partition.py            # extraction-bucket partition
│   ├── arxiv_run_worker.sh           # extraction worker
│   ├── arxiv_aggregate.py            # dedup + OPG-match aggregator
│   ├── arxiv_internal_refs.py        # intra-corpus cross-reference
│   ├── arxiv_review_partition.py     # review-bucket partition
│   ├── arxiv_review_run_worker.sh    # review worker
│   ├── bm_build_records.py           # Bondy–Murty Appendix A records → JSON
│   ├── sync_llm_proof_results.py     # Graph-Theory-LLM-Proofs import
│   ├── sync_openai_math_results.py   # hand-checked OpenAI manuscript matches → JSON
│   ├── conjecture_names.py           # id → canonical name, added to the result files above
│   ├── arxiv_fetch_all.py            # legacy /api/query driver (kept for reference)
│   ├── arxiv_disambig.py             # legacy disambiguation gate
│   ├── status.sh                     # snapshot all OPG workers
│   └── timeout_claude.py             # process-group SIGKILL wrapper for `claude -p`
├── data/
│   ├── problems.json                 # 227 OPG problems
│   ├── reviews/                      # 227 per-slug OPG review JSONs
│   ├── erdos_graph.json              # erdosproblems.com index, 277 problems
│   ├── intersection.json             # fuzzy OPG↔erdos matches; `confirmed` flag per row
│   ├── arxiv_authors.json            # the 12 curated arxiv authors
│   ├── arxiv_conjectures.json        # 768 deduped new conjectures
│   ├── conjecture_names.json         # canonical PREFIX_english_name per conjecture (old id → name)
│   ├── arxiv_extracted/              # 857 raw per-paper extraction outputs
│   ├── arxiv_reviews/                # 762 per-conjecture review JSONs
│   ├── arxiv_opg_matches.{json,tsv}  # arxiv → OPG citation matches (manually triaged)
│   ├── arxiv_internal_refs.{json,tsv} # intra-corpus cross-refs (Phase 1)
│   ├── bondy_murty_conjectures.json  # 38 Bondy–Murty Appendix A items not covered elsewhere
│   ├── bondy_murty_reviews/          # 38 per-item status reviews
│   ├── others_conjectures.json       # hand-written workstream conjectures (Meyniel, …)
│   ├── others_reviews/               # their status reviews
│   ├── llm_proof_results.json        # imported literature corrections and AI write-ups
│   ├── openai_math_results.json      # 22 entries matched to OpenAI manuscripts
│   ├── relations.json                # 225 verified conjecture-to-conjecture relations
│   ├── relations_work/               # relation-pipeline provenance (tags, candidates, verdicts)
│   ├── categories.json
│   └── authors.json
├── problems/                         # self-contained research workstreams
│   ├── directed_path_minimum_outdegree/
│   ├── earth_moon_problem/
│   ├── 3_decomposition_conjecture/
│   ├── pebbling_cartesian_product/
│   ├── unit_vector_flows/
│   └── meyniels_conjecture/
├── PLAN.md                           # OPG crawler / parser / site design
├── LIT_REVIEW.md                     # OPG literature-review design
├── RELATIONS.md                      # conjecture relation graph: pipeline + findings
├── LICENSE                           # MIT for code
└── LICENSE-DATA.md                   # GFDL for data derived from OPG

Featured: directed-path conjecture (Cheng--Keevash Conjecture 1)

A separate workstream attacks Thomassé's directed-path conjecture (every oriented graph with minimum out-degree $\delta$ contains a directed simple path of length $2\delta$), via Cheng--Keevash Lemma 7 and computer-aided enumeration.

The workstream now lives under problems/directed_path_minimum_outdegree/. Its current packaged status closes $\delta \leq 3$, and closes $\delta = 4$ at $n = 9, 10, 11$; the first open local target is $\delta = 4, n \geq 12$.

Licensing summary

• Code: MIT (see LICENSE).
• Data derived from openproblemgarden.org: GFDL v1.2 with attribution (see LICENSE-DATA.md). Each problem record carries a canonical_url field linking back upstream.
• Data derived from arxiv.org: arXiv allows bulk metadata access via OAI-PMH and free distribution under terms documented at info.arxiv.org/help/api/tou.html. Each conjecture record carries the source arxiv_id and abs_url.

Acknowledgments

• The Open Problem Garden community for their decade+ of curating these problems.
• Thomas Bloom for erdosproblems.com, which directly inspired the layout and which we cross-reference.
• The 12 curated arXiv authors whose recent work is mined here.
• Reviews and extraction were generated with Claude (Sonnet 4.6) via claude -p, with every cited URL verified by WebFetch before inclusion. Reviews are advisory and should be spot-checked before being relied upon for research decisions.