Combinatorial geometry and line arrangements

Kobon triangle problem

Collaboration beta

How many nonoverlapping triangular faces can n straight lines create? The packet develops exact wiring and geometric bounds, and reports a sharp 54-triangle result for fourteen lines with at most four finite multiple points. Arrangements with more multiple points remain part of the open problem.

K (n) = \max_{A} T (A) (n \geq 0) .

Known results and sources

Illustration: The three-line example illustrates one bounded triangular face. It does not present a solution for arbitrary numbers of lines.

Research problem

Exact mathematical statement

For an arrangement $A$ of $n$ distinct entire real affine lines, let $T(A)$ count its positive-area bounded triangular faces, with shared boundary edges and vertices permitted and disjoint interiors. Determine, for every integer $n\ge0$ ,

K(n)=\max_{A}T(A).

Parallel lines and finite intersections of arbitrary multiplicity are allowed; coincident lines are not distinct lines. The small values are $K(0)=K(1)=K(2)=0$ . The target requires an unrestricted upper theorem and straight-line constructions attaining it; the source assumes no particular closed formula and leaves the full problem open.

Problem infographic

Problem at a glance

Illustration: A new three-line triangle, permitted parallel and concurrent lines, and a list of counted-face conditions frame the unrestricted Kobon maximum. — Illustration: The unrestricted problem permits parallel lines and multiple intersections and asks for the largest triangular-face count for every n.

Current mathematical picture

Where work on Kobon triangle problem stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureChoose every local pseudoline resolution independently and assume a simultaneous straight realization.

The source’s 36-line Pappus cage has 187 triangular faces and nine triples, with reported straight local optimum 193 versus pseudoline optimum 196. Its conflict graph is edgeless; the obstruction is nonlinear incidence, so the gap is not removed by treating the local choices as independent. The matching formula remains a combinatorial upper bound. Joint full-rank hypotheses give sufficient straight realization, and corank estimates bound the loss; those hypotheses must be checked for the actual arrangement.

Route status · Narrowed route

Main reductionAn exact wiring recurrence bounds straight arrangements

For the generalized wiring maximum W(n), the source gives terminal F_n(w₀,b)=0 and F_n(π,b)=max_legal(g+F_n(π′,b′)), with W(n)=F_n(id,0) and K(n)≤W(n). Equal combinatorial states need not have equal straight realization spaces; the retained implementation only supports 2≤n≤10.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve a uniform Bellman potential on the complete reachable state domain for every n.Task status · Ready to work on

Work mapped so far

Kobon triangle problem in numbers

1.1kretained lines of mathematical investigation1,143 in the current working snapshot

Argument development: 736 · 64%
Explored or eliminated routes: 59 · 5%
Computational analysis: 91 · 8%
Open obligations: 120 · 10%
Definitions and setup: 137 · 12%

9selected mapped statements2routes investigated3open questions3contribution-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.

Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Prove a uniform Bellman potential on the complete reachable state domain for every n.

Suggested move: Specify an invariant state domain and a potential V_n, prove its terminal inequality and its inequality for every legal block event, and evaluate its root value. Preserve the −n term if the proof is expressed through the minimum-cost formulation.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

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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.

Explored alternatives

Other routes

2 recorded

Narrowed routeChoose every local pseudoline resolution independently and assume a simultaneous straight realization.

Route status · Narrowed route

Narrowed routeApply the complete-four-multiple-point theorem to a four-point subgraph of a larger core.

KT-F05 only excludes low-defect arrangements whose full set of finite multiple points has size at most four. The current work explicitly states that extra multiple points change the face and fan constraints, so an overlapping four-point subgraph does not close the five-point frontier. The fixed-core relaxation applies to an explicitly specified complete core. A new five-point classification or a uniform core/fan inequality could extend the bound while preserving all those constraints.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove a uniform Bellman potential on the complete reachable state domain for every n.Suggested move: Specify an invariant state domain and a potential V_n, prove its terminal inequality and its inequality for every legal block event, and evaluate its root value. Preserve the −n term if the proof is expressed through the minimum-cost formulation.

Ready to work on

Control five-or-more-point cores without discarding the extra lines, faces, or fan constraints.Suggested move: For the fourteen-line frontier, classify or safely overrelax all five-point candidates relevant to a 55-triangle arrangement, including parallel and boundary charts. Then identify an argument covering arbitrarily large cores; do not reuse a complete-four-point result on an induced subgraph.

Ready to work on

Construct straight-line arrangements attaining any proposed unrestricted upper bound.Suggested move: Give exact coordinates or a realizability argument for the proposed extremizers. When perturbing multiple points, verify joint incidence and rank conditions, including fixed higher-multiplicity points, and retain all metric alternatives needed by a complete search.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen problem

The problem of determining the maximum number of bounded triangular faces for every number of real affine lines remains open. Published work on simple arrangements and recent preprints provide bounds, constructions, and computational results for particular orders. A result for simple arrangements or pseudolines must retain that restriction when compared with a target allowing parallel lines and multiple intersections.

[1][2][3]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026-04-23
PreprintParpalak and Utkin report a 19-line base arrangement extending an optimal straight-line series at orders 18·2^t+1, using interval and combinatorial checks. This is a restricted infinite family.[3]
2025-07-10
PreprintSavchuk reports a table encoding, SAT searches, and heuristic straightening, including constructions for 23 and 27 lines. Computational exclusions and recovered coordinates have not been replayed here.[2]
2008
Peer reviewedBartholdi, Blanc, and Loisel study triangle maxima in simple affine and projective line and pseudoline arrangements.[1]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction4 of 94
lemma2 of 92
computational claim1 of 91
negative result1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included

retained route statementDetermine K(n)=max_A T(A) for all n≥0, with unrestricted upper bounds and attaining straight-line arrangements.
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementAn exact wiring recurrence bounds straight arrangementsintermediate
retained route statementFinite wiring data and the eight-line attainmentintermediate
retained route statementMatching solves a local combinatorial relaxationintermediate
retained route statementA Pappus cage separates the two local optimaintermediate
retained route statementA complete core controls surviving faces and fansintermediate
retained route statementA sharp fourteen-line result in a restricted classintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureChoose every local pseudoline resolution independently and assume a simultaneous straight realization.reported failure
Useful failureApply the complete-four-multiple-point theorem to a four-point subgraph of a larger core.reported failure
Research targetProve a uniform Bellman potential on the complete reachable state domain for every n.open
Research targetControl five-or-more-point cores without discarding the extra lines, faces, or fan constraints.open
Research targetConstruct straight-line arrangements attaining any proposed unrestricted upper bound.open
ComputationThe source reports W(2),…,W(10)=0,1,2,5,7,11,15,21,25; exhaustive eight-wire classifications and an exact eight-line witness support its K(8)=15 claim. The v8 core calculation has 103,184 distinct surviving-triangle subsets; v9 has 6,362 closed sets across 36 cases and checks its fan recurrence on 5,440 masks.The reported compiled wiring implementation is limited to n≤10. The revision-9 optimizer changes the combinatorial enumeration but shares geometry with revision 8; historical counts of 297,472 visits and 6,582 states were not recovered and are not current output. · reported unreproduced
Narrowed routeChoose every local pseudoline resolution independently and assume a simultaneous straight realization.The source’s 36-line Pappus cage has 187 triangular faces and nine triples, with reported straight local optimum 193 versus pseudoline optimum 196. Its conflict graph is edgeless; the obstruction is nonlinear incidence, so the gap is not removed by treating the local choices as independent. The matching formula remains a combinatorial upper bound. Joint full-rank hypotheses give sufficient straight realization, and corank estimates bound the loss; those hypotheses must be checked for the actual arrangement.
Narrowed routeApply the complete-four-multiple-point theorem to a four-point subgraph of a larger core.KT-F05 only excludes low-defect arrangements whose full set of finite multiple points has size at most four. The current work explicitly states that extra multiple points change the face and fan constraints, so an overlapping four-point subgraph does not close the five-point frontier. The fixed-core relaxation applies to an explicitly specified complete core. A new five-point classification or a uniform core/fan inequality could extend the bound while preserving all those constraints.

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

Priority open bridgeProve a uniform Bellman potential on the complete reachable state domain for every n.

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

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Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve a uniform Bellman potential on the complete reachable state domain for every n.

Kobon triangle problem · ready to start

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Research contextPrepared context for any AI agent

How many nonoverlapping triangular faces can n straight lines create? The current work develops exact wiring and geometric bounds, and reports a sharp 54-triangle result for fourteen lines with at most four finite multiple points. Arrangements with more multiple points remain part of the open problem.

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Current routes and known obstacles
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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
On simple arrangements of lines and pseudo-lines in P^2 and R^2 with the maximum number of trianglespeer reviewed result · Nicolas Bartholdi, Jérémy Blanc, Sébastien Loisel · Contemporary Mathematics 453, 105–116 · 2008 · ARXIV 0706.0723 · DOI 10.1090/conm/453/08797 · accessed Sep 7, 2026
2
Constructing Optimal Kobon Triangle Arrangements via Table Encoding, SAT Solving, and Heuristic Straighteningpreprint · Pavlo Savchuk · arXiv · 2025-07-10 · ARXIV 2507.07951 · accessed Sep 7, 2026
3
The 18·2^t+1 Triangle-Maximal Series of Straight Linespreprint · Roman Parpalak, Denis Utkin · arXiv · 2026-04-23 · ARXIV 2604.22035 · accessed Sep 7, 2026

Important qualifications

This is a bounded external-context collection, not proof checking, an independent mathematical review, legal clearance, or publication authority.
No source programs, datasets, or certificates were executed. A linked resource does not imply reproduction.
No exact checked formalization was confirmed in the scoped search; an empty list does not establish nonexistence.
Unassigned history and recognition fields mean this collection did not establish them. Proposed formal prerequisites are planning tasks, not an exhaustive library inventory.
The earliest original proposal was not inspected; proposer and proposed year remain unassigned.
Recent repository claims concerning individual orders were found but not adjudicated; the collector report records this scope limit. No universal formula or unrestricted order-14 optimum is imported.
MSC 52C30 is drawn from the cited line-arrangement papers. No submitted coordinates or third-party visual assets were used.

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Kobon triangle problem

Exact mathematical statement

Problem at a glance

Where work on Kobon triangle problem stands

Kobon triangle problem in numbers

Prove a uniform Bellman potential on the complete reachable state domain for every n.

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formal and computational footholds

Formalization opportunities

Claims, milestones, and routes in the current map

Conditions that would advance the current route

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Important qualifications

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