This conditional branch begins after the source-reported exclusion of reachable orders at most twelve and asks for compatible degree-six incidence blocks in a six-regular fourteen-vertex complement graph.
Route status · Active routeExtremal combinatorics · hypergraph theory · matchings and transversals
Ryser's Conjecture for Multipartite Hypergraphs
Collaboration betaCan every finite r-partite r-uniform hypergraph be covered by at most r−1 times as many vertices as the number of edges in a largest matching?
Listed inDouglas West's REGS open-problem collectionListed inOpen Problem Garden
Known results and sources 
Research problem
Exact mathematical statement
Let H be a finite r-partite, r-uniform hypergraph, so
for every edge e. Let ν(H) be the largest size of a matching, meaning a family of pairwise disjoint edges, and let τ(H) be the smallest size of a vertex cover, meaning a set of vertices meeting every edge. Ryser's conjecture asks whether
The inequality is known for r=2 and r=3. In general it remains open for every r≥4. The retained handoff develops reductions and conditional branches but explicitly contains no complete proof.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Ryser's Conjecture for Multipartite Hypergraphs stands
The current work develops exact minimal-counterexample reductions and an omission-cylinder reformulation of Ryser's conjecture, proves several conditional closures inside the persistent minimum-pivot branch, records source-attached finite certificates that were not rerun during intake, preserves explicit countermodels to tempting shortcuts, and isolates active order-fourteen, sparse-staircase, and intersecting-design fronts. No complete proof is claimed.
The retained countermodels rule out fixed-matching cover selection, rooted-profile closure without core coupling, two-coordinate petal disjointness, perfect-disjointness, scalar fractional rounding, generic degree edge-coloring, and a frozen-orientation order-twelve step.
Route status · Eliminated routeUnder the explicit persistence hypothesis, reachable varying edges acquire regular colored graph structure, parity, and multiple-intersection constraints.
Evidence posture · Reported reductionThe n=1 case is recast through parallel partitions, strengthened by an edge-count barrier and a punctured-cover trace obstruction, while the general larger-r case stays open.
Evidence posture · Reported special caseClassify six-regular complement graphs on fourteen reachable edges with seven edge-disjoint pivot-color perfect matchings and at least two six-element incidence blocks containing K_6 minus a perfect matching before enumerating full physical partitions.
Task status · Ready to work onWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Reader-facing record corrected; mathematics unchangedWork mapped so far
Ryser's Conjecture for Multipartite Hypergraphs in numbers
1.5kretained lines of mathematical investigation1,537 in the current working snapshot
- Argument development
- 1,280 · 83%
- Explored or eliminated routes
- 80 · 5%
- Computational analysis
- 28 · 2%
- Open obligations
- 20 · 1%
- Definitions and setup
- 129 · 8%
17selected mapped statements7routes investigated8reported milestones8open questions8contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Enumerate the order-fourteen skeleton
Classify six-regular complement graphs on fourteen reachable edges with seven edge-disjoint pivot-color perfect matchings and at least two six-element incidence blocks containing K_6 minus a perfect matching before enumerating full physical partitions.
Suggested move: Enumerate complement graph/block skeletons first and retain exact pivot-color and incidence-block constraints.
What would count as progress
- An exact complete list of admissible six-regular complement graphs and compatible degree-six block systems is produced, or all candidates are excluded.
- Every retained candidate carries seven edge-disjoint pivot-color perfect matchings and the required physical pair coverage constraints.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
Scroll horizontally to explore the route
Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
This conditional branch begins after the source-reported exclusion of reachable orders at most twelve and asks for compatible degree-six incidence blocks in a six-regular fourteen-vertex complement graph.
Route status · Active routeThe exact cylinder theorem and sharp 2r−1 bound reduce the first equality case to a two-coordinate staircase, but physical pivot and blocker consistency remain open.
Route status · Active routeThe n=1 program combines parallel classes, exact degree certificates, punctured-cover trace expansion, and agreement excess without importing the conditional persistence hypothesis.
Route status · Active routeExplored alternatives
Other routes
The current work narrows hard local rank loss to singleton petals and provides recursive trace structure, while the higher-order circuit lemma remains unproved.
Route status · Narrowed routeThe current work reports this conditional branch closed for r≤6 in the stated n ranges and for r=7 through reachable order twelve; it does not close matchings that enter sparse surplus.
Route status · Narrowed routeThe retained countermodels rule out fixed-matching cover selection, rooted-profile closure without core coupling, two-coordinate petal disjointness, perfect-disjointness, scalar fractional rounding, generic degree edge-coloring, and a frozen-orientation order-twelve step.
Route status · Eliminated routeBrowse 1 more explored route
The repaired order-twelve lane requires orbit-complete or all-label enumeration whenever one cover is canonicalized; this remains mandatory for future finite searches.
Route status · Narrowed routeRoute statements and reductions
Statements the next route can inspect and build on
For each maximum matching M, a hypothetical counterexample forces the cylinders defined by all M-sparse edges to cover the entire product of one omitted vertex from each matching edge. Equivalently, the associated finite-domain CNF is unsatisfiable; one uncovered assignment gives an (r−1)n-cover.
Source-reported route statement · dependencies incompleteWhen nu(H)=1 and tau(H)=r, the edges of H are points covered by r parallel classes of incidence blocks; intersecting means every pair of points lies in a common block, while a counterexample requires at least r blocks to cover the point set.
Source-reported route statement · dependencies incompleteFor r≥8, an intersecting r-partite, r-uniform hypergraph with cover number r has at least 3r+1 edges. In an edge-minimal intersecting hypergraph, every nonempty R in a punctured cover induces a trace family on the deleted edge with cover number at least |R|+1.
Source-reported route statement · dependencies incompleteIf sparse cylinders cover the omission product with no complete unary fan, then at least 2r−1 sparse edges are required. Equality forces a two-coordinate staircase: r−1 unary cylinders on one coordinate, r−s unary cylinders on a second, and s residual support-two cylinders.
Source-reported route statement · dependencies incompleteUnder the persistent minimum-branch hypothesis with r=7 and n≥2, the source-reported order-twelve finite certificate plus a written gluing argument imply that the reachable component has at least fourteen edges.
Source-reported route statement · dependencies incompleteIn the conditional r=7 persistent branch at reachable order fourteen, the pivot-graph complement is six-regular, every original vertex has degree at most six, and every six-cover selects at least 2+N_3/2 degree-six incidence blocks; each such block contains K_6 minus a perfect matching.
Source-reported route statement · dependencies incompleteMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Enumerate the order-fourteen skeleton
Classify six-regular complement graphs on fourteen reachable edges with seven edge-disjoint pivot-color perfect matchings and at least two six-element incidence blocks containing K_6 minus a perfect matching before enumerating full physical partitions.
Suggested move: Enumerate complement graph/block skeletons first and retain exact pivot-color and incidence-block constraints.Ready to work on
Close the sparse-surplus staircase physically
Prove that every pivot-realizable equality staircase forces either an augmenting matching or an (r−1)n-cover.
Suggested move: Add pivot and higher-trace blocker consistency to the exact two-coordinate cylinder classification.Ready to work on
Compare blockers across repivots
Track the higher-trace blockers on the almost-complete unary coordinate before and after pivoting through the residual support-two cylinders.
Suggested move: Write the before/after blocker families on the two active staircase coordinates and test which trace constraints survive each repivot.Ready to work on
Prove the intersecting agreement-excess dichotomy
In the n=1 parallel-partition model, prove that small agreement excess forces a Hall-type contradiction or an (r−1)-cover, while large agreement excess compresses repeated intersections to an (r−1)-cover.
Suggested move: Combine punctured-cover trace expansion with the agreement-excess split without assuming persistent minimum behavior.Ready to work on
Prove a structural order-fourteen lemma
Show that two required degree-six incidence blocks force a degree-eight block, a repeated pivot color, or an unavoidable three-set.
Suggested move: Exploit the at-most-two external Q-neighbors of each vertex in a degree-six block before expanding to physical partitions.Ready to work on
Preserve all-orientations search coverage
Every future finite search that canonicalizes one cover must enumerate every orbit of the other omitted-label tuples or all omitted-label choices.
Suggested move: Bind an orbit-completeness proof or explicit all-label enumeration to every persistent-component certificate.Ready to work on
Close singleton-petal circuits
Prove that individually loose singleton petals cannot form higher-order tight deletion circuits across every coordinate unless one of the two-pivot closing alternatives occurs.
Suggested move: Iterate the recursive trace cube created by a tight petal circuit and test it against the two-pivot outcomes.Ready to work on
Prove the two-pivot dichotomy
Find either two disjoint petals avoiding a common maximum matching of G_e or one petal class hit by a minimum cover of G_e.
Suggested move: Analyze exact petal classes together with maximum matchings and minimum covers of the equality core.Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
For finite r-partite r-uniform hypergraphs, the inequality tau(H)<=(r-1)nu(H) remains open for every r>=4. It is proved for r=2 by König's theorem and for r=3 by Aharoni. In the intersecting case nu(H)=1 it is proved through r<=5 but remains open from r=6 onward. Thus the first unresolved general scope is r=4 with nu(H)>=2; weaker bounds, stronger-assumption results, and counterexamples to stronger neighboring conjectures do not settle this scope.
[8][10][12]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintHawranick and Luo resolved a parameter range of a related monochromatic-component covering conjecture for complete multipartite hypergraphs; this does not settle Ryser's matching-versus-transversal inequality.[11] PreprintClow, Haxell, and Mohar disproved a stronger Lovász deletion conjecture for r=3. Their counterexample does not refute Ryser's inequality, which they state remains open for all r>=4.[10] Peer reviewedA peer-reviewed survey recorded the exact general frontier: the conjecture is known for r<=3, and in the intersecting case nu(H)=1 through r<=5.[8] Peer reviewedFrancetić, Herke, McKay, and Wanless proved Ryser's bound for linear intersecting r-partite hypergraphs through r=9; the r=9 proof includes a computation.[6]
Mathematical neighborhood
Related results and reusable starting points
Current focusRyser's conjecture for multipartite hypergraphs
At r=2 the assertion is exactly the matching-cover equality for bipartite graphs supplied by König's theorem.
[8][14]Aharoni's theorem gives tau(H)<=2nu(H) for every tripartite 3-uniform hypergraph.
[2]When nu(H)=1, the hypergraph is intersecting and the conjecture becomes tau(H)<=r-1. This is known for r<=5 and remains open for r>=6.
[3][5]If the hypergraph is both intersecting and linear, the conjectured bound is proved through r=9; the r=9 proof is computational.
[6]The full conjecture is equivalent to covering the vertices of every r-edge-colored graph G by at most (r-1)alpha(G) monochromatic connected subgraphs.
[8]Lovász proposed that r-1 vertices can always be deleted to reduce the matching number. That would imply Ryser by iteration, but a 2025 preprint gives counterexamples already for r=3; Ryser itself survives.
[10]Truncated projective-plane and related constructions attain tau(H)=(r-1)nu(H) for infinite families of uniformities, showing that the factor r-1 cannot be generally improved if the conjecture holds.
[9]A 2026 preprint proves a related covering result for spanning colorings of complete multipartite hypergraphs in a specified parameter range, without proving the Ryser inequality.
[11]Formal and computational footholds
Existing statements, libraries, computations, and datasets that can shorten the next serious attempt.
- computation · source linked; not reproduced by ProofAtlasLinear-intersecting Ryser hypergraph computation
The 2017 study reports exact computational results for linear intersecting multipartite hypergraphs, including a computational r=9 proof and searches/classifications of extremal examples. The paper and arXiv record are linked; ProofAtlas has not rerun or independently checked the computation.
[6][7]
Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetAn aligned Lean statement of the exact r-partite r-uniform inequality was not located in the scoped repositories.
- Formalization targetA formal development would need compatible finite-hypergraph definitions for the partite partition, uniformity, matching number, and transversal number, plus exact statement-alignment review.
Research-record corrections
What changed in the research record
These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.
How the route was assembled
Argument structure
These stages follow the mathematical order of the supplied argument.
8 mapped milestonesretained argument map
Browse all 8 mapped stages
- stage 1Exact critical counterexample model
- stage 2Exact trace and equality-core reductions
- stage 3Exact omission-cylinder reformulation
- stage 4Intersecting design reductions
- stage 5Conditional persistent pivot structure
- stage 6Conditional persistent branch closed through r=6
- stage 7Sharp sparse-surplus staircase
- stage 8Conditional r=7 order-twelve closure
Exact critical counterexample modelThe current work reduces any hypothetical edge-minimal counterexample to an exact one-unit cover violation with strong deletion and finite-kernel constraints.
Research stage 1Mapped research milestoneInitial research sequence
Exact trace and equality-core reductionsThe current work develops exact trace cubes, anti-neighborhood equality cores, deficiency peaks, and singleton-petal rank-loss reductions.
Research stage 2Mapped research milestoneInitial research sequence
Exact omission-cylinder reformulationThe current work reformulates the conjecture exactly as finding an uncovered omission assignment, equivalently satisfying a finite-domain CNF.
Research stage 3Mapped research milestoneInitial research sequence
Intersecting design reductionsThe current work recasts the n=1 case through parallel partitions and adds edge-count and punctured-cover barriers without closing the general intersecting problem.
Research stage 4Mapped research milestoneInitial research sequence
Conditional persistent pivot structureUnder the explicit persistence hypothesis, the current work derives a regular colored pivot graph with parity and multiple-intersection constraints.
Research stage 5Mapped research milestoneInitial research sequence
Conditional persistent branch closed through r=6The current work reports the persistent minimum branch impossible in the stated ranges through r=6, while leaving sparse surplus and larger intersecting ranks open.
Research stage 6Mapped research milestoneInitial research sequence
Sharp sparse-surplus staircaseThe current work proves a sharp 2r−1 sparse-cylinder lower bound and classifies equality as a two-coordinate staircase whose physical closure remains open.
Research stage 7Mapped research milestoneInitial research sequence
Conditional r=7 order-twelve closureThe current work reports a corrected all-orientations finite certificate plus written gluing that pushes the conditional persistent r=7 branch to reachable order at least fourteen.
Research stage 8Mapped research milestoneInitial research sequence
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
16 standing statements1 proposed statements8 mathematical milestones8 open questions3 narrowed routes4 conditional results3 completed special cases
- theorem candidate
1 of 17 1 - reduction
8 of 17 8 - lemma
4 of 17 4 - equivalence
2 of 17 2 - negative result
1 of 17 1 - computational claim
1 of 17 1
Selected mathematical clusters6 mathematical clusters
Exact question and trust boundaryThe r-partite r-uniform tau-versus-nu conjecture, explicit open status, and conditional-persistence scope.1 displayed row
- retained route statementRyser's conjecture for multipartite hypergraphs
Minimal-counterexample reductionsOne-unit criticality, deletion loss, finite kernels, exact traces, equality cores, deficiency peaks, and singleton petals.10 displayed rows · 1 route included
- retained route statementExact edge-critical normalizationintermediate
- retained route statementVertex-deletion loss inequalityintermediate
- retained route statementFinite critical kernelintermediate
- retained route statementExact trace calculusintermediate
- retained route statementEquality core and complete pivot fanintermediate
- retained route statementDeficiency-peak expansionintermediate
- retained route statementSingleton-petal rank-loss reductionintermediate
- Research targetProve the two-pivot dichotomyopen
- Research targetClose singleton-petal circuitsopen
- Narrowed routeSingleton-petal circuit closureThe current work narrows hard local rank loss to singleton petals and provides recursive trace structure, while the higher-order circuit lemma remains unproved.
Cylinder reformulation and sparse surplusExact product-cover/CNF equivalence, sharp sparse witness count, equality staircase, and its open physical closure.6 displayed rows · 1 route included
- retained route statementExact omission-cylinder reformulation
- retained route statementSharp sparse-cylinder surplus and equality staircase
- ChallengeThe product-cylinder equality classification does not by itself realize a physical hypergraph or close Ryser; pivot and higher-trace blocker consistency are still missing.unsupported step · open
- Research targetClose the sparse-surplus staircase physicallyopen
- Research targetCompare blockers across repivotsopen
- Active routeSparse-surplus staircase closureThe exact cylinder theorem and sharp 2r−1 bound reduce the first equality case to a two-coordinate staircase, but physical pivot and blocker consistency remain open.
Conditional persistent-pivot branchPersistent pivot-graph structure, low-rank branch closures, the mixed-evidence r=7 order-twelve closure, and order-fourteen frontier.14 displayed rows · 3 routes included
- retained route statementConditional persistent pivot graphconditional
- retained route statementConditional persistent branch closes through rank sixconditional
- retained route statementConditional order-twelve r=7 closureconditional
- retained route statementOrder-fourteen degree-six block constraintsconditional
- ChallengeThe first order-twelve degree-only search fixed one canonical cover and only one orientation pair, so it was not orientation-complete. The source reports this gap resolved only after an all-orientations physical-partition certificate and separate gluing argument.unsupported step · reported resolved
- ComputationSource-attached dependency-free enumeration of order-twelve degree-four trace multigraphs and compatible seven-color factorization types.The current work reports the exact reduction 3355 labeled trace multigraphs to 24 label orbits, 11 orbits admitting every pivot color separately, and 3 admitting seven edge-disjoint pivot matchings: thin, octet, and nonet. · reported unreproduced
- ComputationSource-attached enumeration of the relative orientations of two thin order-twelve six-covers.The current work reports all 1,920 thin synchronization cases checked, with exactly one survivor at five common-part agreements, and uses this in the no-three-thin-covers argument. · reported unreproduced
- ComputationSource-attached all-orientations and physical-partition certificate for the conditional order-twelve r=7 persistent component.The current work reports 1,024 thin, 128 nonet, and 1,280 octet factorization orbit representatives; 79 compatible three-cover orbit instances; 25 physically realizable orientation signatures; 31 physical partition solutions; and an unavoidable selected triple in every solution. · reported unreproduced
- Research targetEnumerate the order-fourteen skeletonopen
- Research targetProve a structural order-fourteen lemmaopen
- Research targetPreserve all-orientations search coverageopen
- Narrowed routeConditional persistent small-component programThe current work reports this conditional branch closed for r≤6 in the stated n ranges and for r=7 through reachable order twelve; it does not close matchings that enter sparse surplus.
- Active routePersistent r=7 order-fourteen geometryThis conditional branch begins after the source-reported exclusion of reachable orders at most twelve and asks for compatible degree-six incidence blocks in a six-regular fourteen-vertex complement graph.
- Narrowed routeAll-orientations certificate disciplineThe repaired order-twelve lane requires orbit-complete or all-label enumeration whenever one cover is canonicalized; this remains mandatory for future finite searches.
Intersecting n=1 design laneParallel partitions, edge-count and punctured-cover barriers, degree certificates, and the open agreement-excess dichotomy.5 displayed rows · 1 route included
- retained route statementIntersecting parallel-partition reformulationspecial case
- retained route statementIntersecting edge-count and punctured-cover barrierspecial case
- ComputationSource-attached exact integer verifier for the recursive degree-sequence and Caro–Wei edge lower bounds.The current work reports exact lower bounds L_{r,n}(t), including counterexample-threshold tables and intersecting-case values for selected ranks. · reported unreproduced
- Research targetProve the intersecting agreement-excess dichotomyopen
- Active routeIntersecting parallel-partition designThe n=1 program combines parallel classes, exact degree certificates, punctured-cover trace expansion, and agreement excess without importing the conditional persistence hypothesis.
Countermodels and eliminated shortcutsExplicit finite witnesses preserve the precise failure scope of seven tempting approaches.8 displayed rows · 1 route included
- Useful failureDelete one vertex from every edge of a fixed maximum matching and use the retained vertices as a cover.reported failure
- Useful failureDiscard equality-core coupling and infer a hard intersecting subfamily from exact rooted trace-cover identities alone.reported failure
- Useful failureInfer disjoint petals from exact two-coordinate cover number.reported failure
- Useful failureApply perfect-graph machinery to the disjointness graph of a multipartite uniform hypergraph.reported failure
- Useful failureRound the scalar fractional matching lower bound directly to an (n+1)-matching.reported failure
- Useful failureUse maximum vertex degree to force multipartite hypergraph edge-colorability.reported failure
- Useful failureCanonicalize one order-twelve octet cover and freeze one pair of omitted labels for the other two covers.reported failure
- Eliminated routeExplicitly eliminated shortcutsThe retained countermodels rule out fixed-matching cover selection, rooted-profile closure without core coupling, two-coordinate petal disjointness, perfect-disjointness, scalar fractional rounding, generic degree edge-coloring, and a frozen-orientation order-twelve step.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
3 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- An exact complete list of admissible six-regular complement graphs and compatible degree-six block systems is produced, or all candidates are excluded.
- Every retained candidate carries seven edge-disjoint pivot-color perfect matchings and the required physical pair coverage constraints.
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Prepared starting pointEnumerate the order-fourteen skeleton
Ryser's Conjecture for Multipartite Hypergraphs · ready to start
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Research contextPrepared context for any AI agent
Can every finite r-partite r-uniform hypergraph be covered by at most r−1 times as many vertices as the number of edges in a largest matching?
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references18 cited works · next context review by Nov 6, 2026
The mathematical context was checked on Aug 6, 2026. Status can be refreshed sooner after a material result or claim.
- 1Permutation Decompositions of (0,1)-Matrices and Decomposition Transversalsoriginal source · John Robert Henderson · California Institute of Technology · 1971 · DOI 10.7907/J1Z1-SK19 · accessed Aug 6, 2026
- 2Ryser's Conjecture for Tripartite 3-Graphspeer reviewed result · Ron Aharoni · Combinatorica · 2001 · DOI 10.1007/s004930170001 · accessed Aug 6, 2026
- 3Ryser's conjecture on transversals of r-partite hypergraphspeer reviewed result · Zsolt Tuza · Ars Combinatoria 16-B, 201-209 · 1983 · accessed Aug 6, 2026
- 4On Ryser's conjecturepeer reviewed result · P. E. Haxell, A. D. Scott · The Electronic Journal of Combinatorics · 2012-01-21 · DOI 10.37236/1175 · accessed Aug 6, 2026
- 5On Ryser's Conjecture for t-Intersecting and Degree-Bounded Hypergraphspeer reviewed result · Zoltán Király, Lilla Tóthmérész · The Electronic Journal of Combinatorics · 2017-12-22 · ARXIV 1705.10024 · DOI 10.37236/6448 · accessed Aug 6, 2026
- 6On Ryser's Conjecture for Linear Intersecting Multipartite Hypergraphspeer reviewed result · Nevena Francetić, Sarada Herke, Brendan D. McKay, Ian M. Wanless · European Journal of Combinatorics · 2017-03 · ARXIV 1508.00951 · DOI 10.1016/j.ejc.2016.10.004 · accessed Aug 6, 2026
- 7ArXiv record and ancillary material for On Ryser's Conjecture for Linear Intersecting Multipartite Hypergraphssoftware or dataset · Nevena Francetić, Sarada Herke, Brendan D. McKay, Ian M. Wanless · arXiv · 2015; revised 2016 · ARXIV 1508.00951 · accessed Aug 6, 2026
- 8Generalizations and Strengthenings of Ryser's Conjecturesurvey or monograph · Louis DeBiasio, Yigal Kamel, Grace McCourt, Hannah Sheats · The Electronic Journal of Combinatorics · 2021 · ARXIV 2009.07239 · DOI 10.37236/9914 · accessed Aug 6, 2026
- 9A family of extremal hypergraphs for Ryser's conjecturepeer reviewed result · Ahmad Abu-Khazneh, János Barát, Alexey Pokrovskiy, Tibor Szabó · Journal of Combinatorial Theory, Series A · 2019 · ARXIV 1605.06361 · DOI 10.1016/j.jcta.2018.07.011 · accessed Aug 6, 2026
- 10A Counterexample to a Conjecture of Lovászpreprint · Alexander Clow, Penny Haxell, Bojan Mohar · arXiv · 2025-05-08 · ARXIV 2505.05339 · accessed Aug 6, 2026
- 11Covering complete r-partite hypergraphs with few monochromatic componentspreprint · Luke Hawranick, Ruth Luo · arXiv · 2026-03-05 · ARXIV 2603.04704 · accessed Aug 6, 2026
- 12On some special cases of Ryser's conjecture (EGRES TR-2016-14)authoritative webpage · Egerváry Research Group · 2016; page modified 2026-06-25 · accessed Aug 6, 2026
- 13Ryser, Jones, and Tuza Conjecturesmaintained problem list · Douglas B. West's REGS collection · accessed Aug 6, 2026
- 14Ryser's conjecturemaintained problem list · Open Problem Garden · entry posted 2007-03-19 · accessed Aug 6, 2026
- 15Ryser's conjectureencyclopedia · Wikipedia · accessed Aug 6, 2026
- 16List of unsolved problems in mathematicsencyclopedia · Wikipedia · accessed Aug 6, 2026
- 17Formal Conjectures repositoryformalization · The Formal Conjectures Authors · Google DeepMind GitHub repository · accessed Aug 6, 2026
- 18Mathlib.Combinatorics.Hypergraph.Basicformalization · The mathlib Community · mathlib GitHub repository · accessed Aug 6, 2026
Important qualifications
- The Henderson thesis repository page was identified, but its PDF timed out during this run; the exact statement, attribution, and 1971 date were cross-checked against peer-reviewed papers and the REGS entry.
- The original r=4,5 intersecting-case work circulated as a 1978 preprint and appeared as Tuza's 1983 Ars Combinatoria paper; this record uses the published 1983 milestone.
- No exact proof-assistant formalization was found in scoped current Formal Conjectures and mathlib name/path/content searches. Empty formalization results do not establish nonexistence elsewhere.
- The linked 2017 computation was not rerun or independently checked by ProofAtlas; its availability is a literature/resource claim only.
- Searches screened out other statements called Ryser's conjecture, including the Latin-square transversal, design, and circulant-Hadamard problems.
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