[#P2888] Completing a line arrangement to triangular bounded cells

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Problem. Given any finite set \(\mathcal L\) of distinct affine lines in \(\mathbb R^2\), does there exist a finite set \(\mathcal L'\supseteq\mathcal L\) of distinct affine lines such that every bounded connected component of \(\mathbb R^2\setminus\bigcup_{L\in\mathcal L'}L\) is the interior of a triangle? Parallel lines and points incident with three or more lines are allowed in both arrangements.

An illustration accompanying the statement of Completing a line arrangement to triangular bounded cells. — An illustration accompanying the statement. The statement alone defines the problem.

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1Context

Naive repairs of one polygonal cell can create another elsewhere. Recording local completion moves, their newly created cells, and invariant obstructions would make repeated geometric experiments reusable.

2Definitions

Definition 1. A cell of a line arrangement is a connected component of the complement of the union of its lines.

Definition 2. A bounded cell is triangular when its closure is a nondegenerate triangle bounded by segments from exactly three lines of the arrangement.

3What counts as a solution

Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides.
A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has zero answers and records several partial constructions. Current literature searches on simplicial arrangements and arrangement extension did not locate a proof or counterexample for the affine bounded-cell completion problem. Exact unresolved remainder: Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line.^[2][3][4][1]

1Packet records

Show recorded summaries

Record	Kind	Assessment
Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has zero answers and records several partial constructions. Current literature searches on simplicial arrangements and arrangement extension did not locate a proof or counterexample for the affine bounded-cell completion problem. Exact unresolved remainder: Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 Unresolved in this packet after the dated source check. Strongest checked result: The MathOverflow page has zero answers and records several partial constructions. Current literature searches on simplicial arrangements and arrangement extension did not locate a proof or counterexample for the affine bounded-cell completion problem. Exact unresolved remainder: Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line.^[2][3][4][1] Relevance to this problem For Completing a line arrangement to triangular bounded cells, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 2, Reference 3, Reference 4, Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. Strongest checked result: The MathOverflow page has zero answers and records several partial constructions. Current literature searches on simplicial arrangements and arrangement extension did not locate a proof or counterexample for the affine bounded-cell completion problem. Exact unresolved remainder: Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line. Open full record →
Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line.^[1] Relevance to this problem For line arrangement triangular bounded completion, sets the next completion test: Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Work against the displayed statement and preserve every hypothesis and quantifier. Prove a finite augmentation procedure for every initial arrangement, or give a finite initial arrangement and prove that every finite line extension leaves a bounded cell with at least four sides. A negative result may use a projective or incidence invariant, but it must show that the invariant survives every allowed added affine line. Any computation must retain a replayable witness and a matching exclusion or completeness certificate. Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-27. The MathOverflow page has zero answers and records several partial constructions. Current literature searches on simplicial arrangements and arrangement extension did not locate a proof or counterexample for the affine bounded-cell completion problem.

On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for question 155196; all comments were checked, including the quadrilateral completion and vertical-line partial results.
Cuntz, arXiv:1108.3000, classifies projective simplicial arrangements through 27 lines. That finite classification concerns arrangements already simplicial and does not decide completion of an arbitrary affine subarrangement.
Geis, arXiv:1809.09362, gives combinatorial restrictions for simplicial arrangements but no arbitrary completion theorem was found in the dated search.
Cuntz, arXiv:2607.17785, studies simplicial arrangements with special vertices. The July 2026 paper does not supply an arbitrary affine completion theorem.
A TheoremDB search for triangulating line arrangements, simplicial completion, and triangular bounded cells found no duplicate.
Independent source, duplicate, exact-title, and equivalent-formulation review completed on 2026-08-01.

Recorded example 1. Adding every line through a generic point and an old intersection ensures that each bounded cell has at most four sides; this known partial construction stops one side short of the target.

2See also

Grid points with no three collinear and no four concyclicdiscrete geometry
Largest equilateral-triangle-free subset of the fifteen-row triangular latticediscrete geometry
Most lattice points with all pairwise slopes distinct in a ten by ten griddiscrete geometry

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Plain text

“Completing a line arrangement to triangular bounded cells.” TheoremDB. P2888. Problem statement; statement text SHA-256 1020feae073191ebc6c5708f757380a597c3fe57d15c8b3701a9d73668c25215. https://theoremdb.org/statement/?ref=P2888

BibTeX

@misc{theoremdb-problem-1020feae073191ebc6c5708f757380a597c3fe57d15c8b3701a9d73668c25215,
  title = {{Completing a line arrangement to triangular bounded cells}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 1020feae073191ebc6c5708f757380a597c3fe57d15c8b3701a9d73668c25215},
  url = {https://theoremdb.org/statement/?ref=P2888}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from mathoverflow.net^[1], current as of August 1, 2026.

1References

Packet source. MathOverflow question 155196, “Completing a line arrangement to triangular bounded cells,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗forum · reference source · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at Question 155196 and all visible comments, checked through the Stack Exchange API on 2026-07-27.Also cited at Full question, answers, and visible comments concerning Completing a line arrangement to triangular bounded cells; checked 2026-08-01.Also cited at Editorial research route recorded 2026-08-01.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Completing a line arrangement to triangular bounded cells: This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.Source named by the research packet.
arXiv preprint 1108.3000, linked primary source for “Completing a line arrangement to triangular bounded cells,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:1108.3000, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and exhaustive classification of real-projective simplicial arrangements with at most 27 lines.Source used to assess the problem's recorded status.For Completing a line arrangement to triangular bounded cells, this source classifies arrangements that are already simplicial and does not show that every affine arrangement admits a triangular completion.
arXiv preprint 1809.09362, linked primary source for “Completing a line arrangement to triangular bounded cells,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:1809.09362, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and t-vector inequalities and finiteness results for free and simplicial real-projective pseudoline arrangements.Source used to assess the problem's recorded status.For Completing a line arrangement to triangular bounded cells, this source supplies structural constraints on simplicial arrangements rather than a universal affine completion procedure.
arXiv preprint 2607.17785, linked primary source for “Completing a line arrangement to triangular bounded cells,” checked 2026-08-01. Question statement, visible answers and comments, or the linked article sections described in the source record. ↗preprint · primary source · arXiv:2607.17785, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Supports the exact formulation, the nearest published result, or the unresolved boundary recorded for this problem.Also cited at abstract and main bound of at most 12 parallel classes for the studied arrangements with a special vertex.Source used to assess the problem's recorded status.For Completing a line arrangement to triangular bounded cells, this source treats a recent special family and neither proves nor refutes completion of an arbitrary initial affine arrangement.

This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.

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[#P2888] Completing a line arrangement to triangular bounded cells

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