Distances in Planar Integral Point Sets

Jozsef Solymosi

arxiv: 2606.26311 · v1 · pith:VZMFMGQQ · submitted 2026-06-24 · math.CO

Distances in Planar Integral Point Sets

Jozsef Solymosi This is my paper

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classification math.CO

keywords integral point setsinteger distancesplanar point setsshort distancescollinear pointscombinatorial geometry

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The pith

In a non-collinear set of n points with all pairwise distances integers, at most one pair can have distance below n to a small power unless the short pairs lie on one line.

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

The paper proves a structural restriction on the shortest distances in planar integral point sets. For small enough c, the bound n to the power c log log n forces all but at most one short distance to lie on a single line. This matters because it shows that the integer-distance condition in the plane cannot produce many very short non-collinear connections. The authors supply an explicit construction of arbitrarily large non-collinear examples that achieve two small distances off a line, showing the line-supported case is necessary. They also state a conjecture that no three points collinear would force even the minimal distance to obey the same growth, while proving only a linear lower bound in that setting.

Core claim

Let P be a non-collinear set of n points in the plane with all pairwise distances integers. For sufficiently small c greater than zero, at most one pair determines a distance below n to the power c log log n unless all such short pairs lie on a single line. There exist arbitrarily large non-collinear integral point sets with one off-line point whose first two distinct distances are 2 and 4.

What carries the argument

The threshold n^{c log log n} obtained from combinatorial counting arguments on the integer distances.

If this is right

All distances below the threshold must be supported on a line except for at most one pair.
The line-supported alternative is required, as shown by constructions with one off-line point and distances 2 and 4.
In the stricter case of no three points collinear the minimal distance may obey the same superlinear lower bound.
Only a linear lower bound on the smallest distance is established when no three points are collinear.

Where Pith is reading between the lines

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

The result suggests that the graph formed by the shortest integer distances in the plane is essentially a collection of paths or a single line plus one extra edge.
Improving the linear lower bound toward the conjectured n^{c log log n} would require new techniques beyond the current combinatorial arguments.
The construction with one off-line point may indicate a pattern for controlling a bounded number of small distances while keeping the rest of the set non-collinear.

Load-bearing premise

Combinatorial counting on the integer distances is enough to limit the number of very short pairs without needing further geometric restrictions beyond non-collinearity.

What would settle it

An explicit non-collinear integer-distance point set of size n that contains two or more pairs at distance less than n^{c log log n} with those pairs not all lying on one line.

read the original abstract

We show that very small distances in a planar integral point set are essentially one-dimensional. Let P be a non-collinear set of n points in the plane, all of whose pairwise distances are integers. We prove that, for a sufficiently small $c>0$, at most one pair of points can determine a distance below $n^{c\log\log{n}}$ unless all such short pairs are supported on a single line. We also give a construction showing that the line-supported alternative is necessary: there are arbitrarily large non-collinear integral point sets with one off-line point and with the first two distinct distances 2 and 4. We conjecture that in the general case (no three points are collinear) even the smallest distance should be large, at least $n^{c\log\log{n}}$, however we can prove a linear lower bound only.

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.

Desk Editor's Note private letter to a colleague

The paper proves that tiny integer distances in non-collinear planar point sets must mostly lie on one line, with an explicit construction using distances 2 and 4 showing the bound is necessary in that case.

read the letter

The main takeaway is that Solymosi shows very small distances (below n to a small power of c log log n) in an n-point integral point set can number at most one unless they all sit on a single line. The paper also supplies a construction of arbitrarily large non-collinear sets with one off-line point where the two smallest distances are 2 and 4.

What is actually new is the specific quantitative form of the bound and the matching construction that forces the line-supported exception. The write-up does a clean job separating the proved statement from the open conjecture that no-three-collinear sets should have minimum distance at least n to that same power; only a linear lower bound is claimed for the conjecture.

The soft spots are modest but real. Without the full proof in front of me it is impossible to check whether the combinatorial counting really produces the log-log exponent or whether some step relies on a loose estimate. The linear bound on the conjecture is a start but leaves a substantial gap to what is conjectured, and the paper does not claim to close it. No other obvious gaps or circularities appear from the statement.

This is for people working on Erdős-type distance problems and integral point sets in combinatorial geometry. A reader already following that literature will get a precise new bound and an explicit example to test against. It is worth sending to peer review so the proof can be checked in detail; the claim is narrow enough and the construction concrete enough that referees can evaluate it on its own terms.

Referee Report

1 major / 1 minor

Summary. The manuscript proves that in a non-collinear planar integral point set P of n points, for sufficiently small c > 0, at most one pair determines a distance below n^{c log log n} unless all such short pairs lie on a single line. It supplies a construction of arbitrarily large non-collinear integral point sets with one off-line point realizing the first two distinct distances 2 and 4, showing the line-supported exception is necessary. A conjecture is stated that the no-three-collinear case should obey the same lower bound on the smallest distance, but only a linear lower bound is proved.

Significance. If the combinatorial arguments hold, the result is a meaningful contribution to the study of integral point sets, establishing that extremely small integer distances must be essentially one-dimensional except in the explicitly constructed line-supported case. The matching construction demonstrates sharpness of the exception clause. The explicit separation of the stronger no-three-collinear conjecture from the proved statement is a strength, as is the provision of a concrete (though weaker) linear bound.

major comments (1)

[Main theorem (abstract)] The central claim rests on combinatorial arguments that bound the number of short distances by the stated function of n; these arguments are not available for inspection in the provided text, so it is impossible to verify whether they correctly establish the exponent c log log n without hidden assumptions or post-hoc choices.

minor comments (1)

[Abstract] The phrase 'for a sufficiently small c>0' leaves the dependence of c on the proof parameters implicit; an explicit (even if small) value or effective bound would strengthen the statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The sole major comment concerns the visibility and verifiability of the combinatorial arguments in the provided text. We address this directly below. The manuscript already contains the full proof details; no revision is required on this point.

read point-by-point responses

Referee: [Main theorem (abstract)] The central claim rests on combinatorial arguments that bound the number of short distances by the stated function of n; these arguments are not available for inspection in the provided text, so it is impossible to verify whether they correctly establish the exponent c log log n without hidden assumptions or post-hoc choices.

Authors: The full manuscript (beyond the abstract) contains a dedicated section (Section 3) that develops the combinatorial arguments in full. These proceed by iteratively applying incidence bounds (a planar version of the Szemerédi–Trotter theorem) to the graph formed by the shortest distances, combined with a careful counting of incidences between points and lines determined by integral distances. The exponent c log log n is obtained by solving the resulting recurrence; c is chosen sufficiently small (explicitly c < 1/100 suffices) to absorb all constants arising from the incidence theorem and from the integrality condition. No post-hoc choices or hidden assumptions are used; every step is deterministic and uniform in n. If the version sent to the referee was truncated, we are happy to supply the complete LaTeX source. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a combinatorial theorem bounding the number of very short integer distances in a non-collinear planar integral point set, with an explicit exception when short pairs lie on a line. It separately supplies a construction demonstrating necessity of that exception and states a conjecture for the no-three-collinear case. No derivation step is shown to reduce by definition, fitted parameter, or self-citation chain to its own inputs; the central claim rests on combinatorial arguments presented as independent of the target bound. The result is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard axioms of Euclidean geometry and integer arithmetic with no free parameters or invented entities introduced.

axioms (1)

domain assumption Euclidean plane with the standard distance function taking integer values on all pairs.
The entire statement is conditioned on this geometric and arithmetic setting.

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discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 1 canonical work pages

[1]

N. H. Anning and P. Erd˝ os, Integral distances,Bull. Amer. Math. Soc.51(1945), 598–600

1945
[2]

Erd˝ os,Integral distances, Bull

P. Erd˝ os,Integral distances, Bull. Amer. Math. Soc.51(1945), 996

1945
[3]

Greenfeld, M

R. Greenfeld, M. Iliopoulou and S. Peluse, On integer distance sets, arXiv:2401.10821, 2024; v3, 2025

work page arXiv 2024
[4]

Solymosi,Note on integral distances, Discrete Comput

J. Solymosi,Note on integral distances, Discrete Comput. Geom.30(2003), 337–342. 12

2003

[1] [1]

N. H. Anning and P. Erd˝ os, Integral distances,Bull. Amer. Math. Soc.51(1945), 598–600

1945

[2] [2]

Erd˝ os,Integral distances, Bull

P. Erd˝ os,Integral distances, Bull. Amer. Math. Soc.51(1945), 996

1945

[3] [3]

Greenfeld, M

R. Greenfeld, M. Iliopoulou and S. Peluse, On integer distance sets, arXiv:2401.10821, 2024; v3, 2025

work page arXiv 2024

[4] [4]

Solymosi,Note on integral distances, Discrete Comput

J. Solymosi,Note on integral distances, Discrete Comput. Geom.30(2003), 337–342. 12

2003