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arxiv: 1907.09331 · v1 · pith:IRVRG64Gnew · submitted 2019-07-22 · 🧮 math.CO

On diameter bounds for planar integral point sets in semi-general position

N.N. Avdeev This is my paper

Pith reviewed 2026-05-24 18:05 UTC · model grok-4.3

classification 🧮 math.CO
keywords planar integral point setssemi-general positiondiameter boundsinteger distancescombinatorial geometryno three collinear
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The pith

The minimum diameter of planar integral point sets in semi-general position grows faster than linearly with the number of points.

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

The paper proves a new lower bound on the minimum diameter of planar integral point sets in semi-general position that improves on the existing linear bound. These sets consist of non-collinear points in the plane whose pairwise distances are all integers, with the added condition that no three points are collinear. A reader would care because the bound shows that avoiding collinear triples while keeping distances integral forces the points to spread out more than linear growth would allow. The argument works directly in the combinatorial setting of such finite sets of fixed cardinality.

Core claim

We prove a new lower bound for minimum diameter of planar integral point sets in semi-general position that is better than linear.

What carries the argument

The minimum diameter taken over all n-point planar integral point sets in semi-general position, shown to be superlinear in n.

Load-bearing premise

The minimum diameter is taken over all finite planar integral point sets of a given cardinality in semi-general position.

What would settle it

An explicit construction of an n-point planar integral point set in semi-general position whose diameter is at most linear in n would falsify the claim.

read the original abstract

A point set $M$ in the Euclidean plane is called a planar integral point set if all the distances between the elements of $M$ are integers, and $M$ is not situated on a straight line. A planar integral point set is called to be in semi-general position, if it does not contain collinear triples. The existing lower bound for mininum diameter of planar integral point sets is linear. We prove a new lower bound for mininum diameter of planar integral point sets in semi-general position that is better than linear.

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 / 0 minor

Summary. The paper claims to prove a new lower bound on the minimum diameter of finite planar integral point sets (all pairwise distances integers, not collinear) in semi-general position (no three points collinear) that improves on the known linear bound and is superlinear in the cardinality n.

Significance. If the claimed combinatorial argument holds and yields a verifiable superlinear bound without hidden parameters or reductions to the linear case, the result would strengthen known constraints on integer-distance realizations in the plane under the no-three-collinear condition. The setting matches the standard definition of the problem exactly.

major comments (2)
  1. [Whole manuscript] The manuscript consists solely of the abstract; no proof, combinatorial counting argument, or derivation is supplied. This makes it impossible to verify whether the transition from the linear bound respects the semi-general position condition or contains gaps.
  2. [Abstract] No explicit statement of the new bound (e.g., Ω(n^{1+ε}) for some ε>0, or a concrete function of n) appears, preventing assessment of whether it is load-bearing or merely asymptotic.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for these comments. The submission appears to have included only the abstract, which we will correct by providing the full manuscript containing the combinatorial argument. We address each point below.

read point-by-point responses

  1. Referee: [Whole manuscript] The manuscript consists solely of the abstract; no proof, combinatorial counting argument, or derivation is supplied. This makes it impossible to verify whether the transition from the linear bound respects the semi-general position condition or contains gaps.

    Authors: The full manuscript contains a combinatorial counting argument that derives the superlinear lower bound on the diameter while enforcing the semi-general position condition (no three points collinear). We will resubmit the complete version with the full derivation so that the transition from the known linear bound and any potential gaps can be directly verified. revision: yes

  2. Referee: [Abstract] No explicit statement of the new bound (e.g., Ω(n^{1+ε}) for some ε>0, or a concrete function of n) appears, preventing assessment of whether it is load-bearing or merely asymptotic.

    Authors: We agree that the abstract should state the bound explicitly rather than only describing it as 'better than linear.' The argument yields a concrete superlinear function of n; we will revise the abstract to include the precise asymptotic form. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a combinatorial proof establishing a superlinear lower bound on the minimum diameter of n-point planar integral point sets in semi-general position (no three collinear). The abstract and skeptic analysis indicate that the derivation relies on counting arguments respecting the no-three-collinear condition and matches the stated definitions exactly, without reducing any prediction or bound to a fitted parameter, self-referential definition, or load-bearing self-citation chain. No equations or steps are described that equate the claimed result to its inputs by construction. The setting is self-contained as a direct proof within the combinatorial framework, yielding an independent mathematical result rather than a renaming or imported uniqueness claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions of integral point sets and semi-general position; no free parameters, new entities, or non-standard axioms are visible in the abstract.

axioms (2)
  • domain assumption All pairwise distances are integers and the set is not collinear.
    This is the definition of a planar integral point set given in the abstract.
  • domain assumption No three points are collinear.
    This is the definition of semi-general position stated in the abstract.

pith-pipeline@v0.9.0 · 5604 in / 1166 out tokens · 22865 ms · 2026-05-24T18:05:03.442694+00:00 · methodology

discussion (0)

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Reference graph

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