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arxiv: 2604.22035 · v1 · submitted 2026-04-23 · 🧮 math.CO

The 18cdot 2^t+1 Triangle-Maximal Series of Straight Lines

Roman Parpalak , Denis Utkin This is my paper

Pith reviewed 2026-05-09 20:46 UTC · model grok-4.3

classification 🧮 math.CO
keywords line arrangementsbounded trianglesiterative constructioncombinatorial geometrymaximum trianglesaffine arrangementscomputer-assisted verification
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The pith

A 19-line arrangement seeds an infinite family of line arrangements with the maximum number of bounded triangles for every n of the form 18·2^t + 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs one specific arrangement of 19 straight lines in the affine plane that meets all conditions required by a known iterative construction for adding lines while keeping the triangle count maximal. Applying the iteration repeatedly to this base produces optimal arrangements at every size n = 18·2^t + 1. Computer-assisted interval arithmetic and combinatorial checks confirm that the 19-line case satisfies the necessary geometric and incidence conditions for unlimited repetition. Separate exhaustive searches for 21, 23 and 27 lines find no other bases that support the same full iteration, yet identify two larger configurations reachable by a single iteration step that still match the known upper bound.

Core claim

We construct a straight-line affine arrangement of 19 lines satisfying the conditions of the iterative construction by Bartholdi, Blanc, and Loisel, thereby obtaining an infinite series of straight-line arrangements attaining the maximum number of bounded triangles for every n=18·2^t+1. The conditions are verified by computer-assisted interval and combinatorial checks. A computational search over n=21, 23, 27 lines provides strong evidence against the existence of further base configurations compatible with the known iterative constructions, but reveals arrangements allowing a single iterative step that yield arrangements of 41 and 45 lines with 533 and 645 bounded triangles, respectively, 5

What carries the argument

The Bartholdi-Blanc-Loisel iterative construction applied to a verified 19-line base arrangement that preserves maximality of bounded triangular faces under repeated doubling steps.

If this is right

  • Arrangements with n = 18·2^t + 1 lines attain the maximum possible number of bounded triangular faces for every nonnegative integer t.
  • The same maximum is realized by straight-line affine arrangements rather than pseudolines.
  • Two single-iteration extensions produce 41-line and 45-line arrangements that also match the upper bound with 533 and 645 triangles respectively.
  • No additional base configurations compatible with the full iteration exist for line counts between 21 and 27.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The 19-line configuration appears to be the unique small seed that permits unlimited iteration under the known rules.
  • If no other iterative schemes exist, the construction may settle the exact maximum for this arithmetic sequence of n.
  • The computer search results suggest that partial extensions (single steps) can still reach the bound even when full iteration is impossible.

Load-bearing premise

The 19-line arrangement satisfies every geometric and combinatorial condition required by the Bartholdi-Blanc-Loisel iterative construction.

What would settle it

An explicit counterexample arrangement with n=37 lines that has fewer bounded triangles than the number predicted by one iteration from the 19-line base, or a manual verification that the 19-line arrangement violates at least one of the iterative conditions.

Figures

Figures reproduced from arXiv: 2604.22035 by Denis Utkin, Roman Parpalak.

Figure 1
Figure 1. Figure 1: The base configuration for n = 19 straight lines [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A simple arrangement of four lines, its O-matrix, and the correspondence between faces and adjacency in the matrix. Lemma 3.1 (O-matrix criterion for bounded faces). Let (Li1 , . . . , Lik ) be a cycli￾cally ordered list of distinct pseudolines in a simple arrangement. Then these pseu￾dolines bound a bounded k-gonal face if and only if for every s = 1, . . . , k (with i0 = ik and ik+1 = i1) the two labels … view at source ↗
Figure 3
Figure 3. Figure 3: Pseudoline diagram of the base configuration with all 107 bounded triangles numbered. Lines are labeled 0, 1, . . . , 18. [ [1, 11, 3, 7, 2, 12, 9, 17, 8, 15, 5, 13, 10, 14, 6, 16, 4, 18], [0, 11, 17, 7, 15, 12, 18, 13, 14, 9, 16, 8, 10, 5, 6, 3, 4, 2], [3, 11, 7, 0, 12, 17, 9, 15, 8, 13, 5, 14, 10, 16, 6, 18, 4, 1], [2, 11, 0, 7, 17, 12, 15, 9, 13, 8, 14, 5, 16, 10, 18, 6, 1, 4], [5, 7, 6, 9, 8, 11, 10, 1… view at source ↗
Figure 4
Figure 4. Figure 4: A realizable 41-line arrangement with 533 bounded triangles, matching the upper bound ⌊n(n − 2)/3⌋, obtained from a 21-line arrangement after one iterative step of [3] [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: A realizable 45-line arrangement with 645 bounded triangles, matching the upper bound ⌊n(n − 2)/3⌋, obtained from a 23-line arrangement after one iterative step of [3] [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A realizable 49-line arrangement with 766 bounded triangles, obtained from a 25-line arrangement after one iterative step of [3]. After adding the line at infinity, one obtains a projective arrangement of 50 lines with 815 triangles, attaining the upper bound (7.1) [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
read the original abstract

Given $n$ lines in general position in the plane, how many bounded triangular faces can the arrangement have? We construct a straight-line affine arrangement of $19$ lines satisfying the conditions of the iterative construction by Bartholdi, Blanc, and Loisel, thereby obtaining an infinite series of straight-line arrangements attaining the maximum number of bounded triangles for every $n=18\cdot 2^t+1$. The conditions are verified by computer-assisted interval and combinatorial checks. A computational search over $n=21$, $23$, $27$ lines provides strong evidence against the existence of further base configurations compatible with the known iterative constructions, but reveals arrangements allowing a single iterative step that yield arrangements of $41$ and $45$ lines with $533$ and $645$ bounded triangles, respectively, each matching the upper bound.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper constructs a specific straight-line arrangement of 19 lines in the plane that satisfies the iterative conditions of Bartholdi, Blanc, and Loisel, yielding an infinite family of arrangements attaining the maximum number of bounded triangles for every n = 18·2^t + 1. It reports computer-assisted interval and combinatorial verification of the base case together with exhaustive searches over n = 21, 23, 27 that find no further base configurations but identify single-iteration extensions to 41- and 45-line arrangements achieving 533 and 645 bounded triangles, respectively.

Significance. If the 19-line base case is correct, the result supplies the first explicit infinite straight-line family achieving the known upper bound on bounded triangles for an arithmetic progression of n, thereby confirming that the BBL iteration can be realized geometrically without loss of maximality. The additional computational search data strengthens the claim that no other small base cases exist for the known iterations and supplies concrete new examples (41 and 45 lines) that attain the bound after one step.

major comments (2)
  1. [§3] §3 (19-line base configuration): the assertion that the arrangement satisfies every prerequisite of the BBL iteration (general position, exact crossing pattern, no unintended parallels or concurrencies) rests solely on high-level computer-assisted interval and combinatorial checks; the manuscript supplies neither the explicit list of predicates verified, the numerical coordinates of the lines, floating-point error bounds, nor the verification code or raw output.
  2. [§4] §4 (computational search): the claim that the 41- and 45-line arrangements achieve exactly 533 and 645 bounded triangles and thereby match the upper bound is stated without an explicit reference to the formula or derivation of that bound in the same section, making independent verification of the matching claim impossible from the given data.
minor comments (1)
  1. [Abstract] The abstract mentions 'the conditions' of the BBL construction without a brief parenthetical reminder of what those conditions are, which would aid readers unfamiliar with the reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback. We address the major comments below and will make the necessary revisions to the manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (19-line base configuration): the assertion that the arrangement satisfies every prerequisite of the BBL iteration (general position, exact crossing pattern, no unintended parallels or concurrencies) rests solely on high-level computer-assisted interval and combinatorial checks; the manuscript supplies neither the explicit list of predicates verified, the numerical coordinates of the lines, floating-point error bounds, nor the verification code or raw output.

    Authors: We agree with the referee that the verification details are insufficiently documented in the current manuscript. To address this, we will expand §3 to include the explicit list of predicates verified by the computer-assisted checks, the numerical coordinates of the 19 lines (with appropriate precision), the floating-point error bounds, and a description of the combinatorial verification. Additionally, we will provide the verification code and raw output as supplementary material. This will allow readers to independently confirm that the arrangement satisfies all prerequisites of the BBL iteration. revision: yes

  2. Referee: [§4] §4 (computational search): the claim that the 41- and 45-line arrangements achieve exactly 533 and 645 bounded triangles and thereby match the upper bound is stated without an explicit reference to the formula or derivation of that bound in the same section, making independent verification of the matching claim impossible from the given data.

    Authors: We acknowledge that §4 would be improved by an explicit reference to the upper bound formula. In the revised version, we will include the formula for the maximum number of bounded triangles in an arrangement of n lines (as derived in the work of Bartholdi, Blanc, and Loisel), and explicitly compute or reference the values for n=41 and n=45 to show that 533 and 645 match the bound. This will make the matching claim verifiable from the section. revision: yes

Circularity Check

0 steps flagged

No circularity; concrete base case verified independently of iteration

full rationale

The derivation constructs an explicit 19-line straight-line arrangement and asserts that computer-assisted interval and combinatorial checks confirm it meets the prerequisites of the external Bartholdi-Blanc-Loisel iteration. This base configuration is a distinct geometric object whose properties are checked against external criteria rather than being defined in terms of the iterated output or fitted to the target triangle counts. The infinite series is then obtained by applying the cited external construction, with no self-definitional reduction, no fitted parameter renamed as prediction, and no load-bearing self-citation chain. The central claim therefore remains self-contained against the provided verification step.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard axioms of Euclidean plane geometry and the correctness of the cited iterative construction; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (1)
  • domain assumption The Bartholdi-Blanc-Loisel iterative conditions are correctly stated and applicable to straight-line arrangements.
    Invoked in the abstract as the basis for the infinite series.

pith-pipeline@v0.9.0 · 5438 in / 1226 out tokens · 27653 ms · 2026-05-09T20:46:37.273248+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

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