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arxiv: 2606.05841 · v1 · pith:SBAXSZUGnew · submitted 2026-06-04 · 🧮 math.CO

Geometric Sidon Problems

Felix Christian Clemen , Jakob F\"uhrer , Oliver Roche-Newton This is my paper

Pith reviewed 2026-06-28 00:52 UTC · model grok-4.3

classification 🧮 math.CO
keywords geometric Sidon setsdistinct distancespoint sets in the planehypergraph independenceincidence geometryextremal combinatoricscombinatorial geometry
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The pith

Any finite point set in the plane contains a subset of size at least the cube root of the total with all pairwise distances distinct.

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

The paper proves that from any finite collection of points in the plane one can always extract a subset whose size is at least on the order of the cube root of the original set, and in which every pair determines a unique distance. The argument models repeated distances as hyperedges in an auxiliary hypergraph and invokes a general theorem on the independence number of hypergraphs whose edges are suitably distributed; incidence geometry supplies the needed distribution bounds. A sympathetic reader would care because the result gives a quantitative guarantee on how large a subset can be while avoiding any repeated distance, improving the exponent in a prior theorem of Charalambides.

Core claim

For any finite point set P subset of R^2 there exists a subset P' subset of P with |P'| much larger than |P| to the power 1/3 such that all distances determined by pairs in P' are distinct. The proof proceeds by constructing a 4-uniform hypergraph whose vertices are the pairs from P and whose hyperedges correspond to pairs of pairs that realize the same distance, then applying the Li-Postle theorem on hypergraph independence numbers once suitable edge-distribution conditions are verified via incidence geometry.

What carries the argument

The 4-uniform hypergraph on the edges of the complete graph on P whose hyperedges link pairs of pairs that determine the same distance; the Li-Postle independence-number bound is applied once the edge-distribution hypotheses are confirmed by incidence estimates.

If this is right

  • Every finite point set in the plane admits a distinct-distance subset of size Omega(n^{1/3}).
  • The exponent 1/3 improves the earlier bound obtained by Charalambides.
  • The argument applies to arbitrary finite point sets without further restrictions on their configuration.
  • Incidence geometry supplies the edge-distribution estimates needed to invoke the hypergraph theorem.

Where Pith is reading between the lines

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

  • The same hypergraph-plus-incidence approach may extend to other forbidden geometric configurations such as repeated angles or isosceles triangles.
  • Refinements of the incidence bounds could potentially raise the exponent above 1/3.
  • The method may adapt to point sets in higher dimensions or under different distance functions.

Load-bearing premise

The hypergraph whose hyperedges correspond to pairs of pairs sharing a distance satisfies the specific edge-distribution hypotheses required by the Li-Postle theorem.

What would settle it

A finite point set P in the plane for which every subset with all pairwise distances distinct has size o(|P|^{1/3}).

Figures

Figures reproduced from arXiv: 2606.05841 by Felix Christian Clemen, Jakob F\"uhrer, Oliver Roche-Newton.

Figure 1
Figure 1. Figure 1: A rhombus with center m and diagonals ℓ1, ℓ2. passing through at least two points of P and let D := {(ℓ1, ℓ2) ∈ L 2 | ℓ1 ⊥ ℓ2 ∧ |ℓ1 ∩ P| ≥ |ℓ2 ∩ P|}. Further, let R(P) denote the number of rhombi defined by P. For a line ℓ and point x, write rℓ(x) := |{{u, v} ∈ ℓ∩P 2  | u + v = 2x}|. For two distinct non parallel lines ℓ1 and ℓ2, let mℓ1,ℓ2 denote the point of intersection of the two lines. Then R(P) ≤ X … view at source ↗
read the original abstract

This paper considers geometric problems of the following type: given a point set $P \subset \mathbb R^2$, one seeks a large subset avoiding a prescribed geometric configuration. Our main result states that, for any $P \subset \mathbb R^2$, there exists a subset $P' \subset P$ with $|P'| \gg |P|^{1/3}$ such that all of the distances determined by $P'$ are distinct. This improves a result of Charalambides. We make heavy use of a result of Li and Postle concerning the independence number of hypergraphs which satisfy some edge distribution conditions, as well as tools from incidence geometry.

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 / 2 minor

Summary. The paper proves that for any finite point set P ⊂ R² there exists P' ⊂ P with |P'| ≫ |P|^{1/3} such that all pairwise distances realized by P' are distinct. The argument models the problem via a 4-uniform hypergraph H whose vertices are pairs from P and whose edges encode pairs of pairs that determine the same distance; incidence-geometry bounds are used to verify that H satisfies the edge-distribution hypotheses of the Li–Postle theorem, which then supplies the claimed independence-number lower bound. The result improves the exponent obtained by Charalambides.

Significance. If the verification that the geometric hypergraph meets the Li–Postle conditions holds without hidden losses, the work gives a concrete new application of hypergraph independence theorems to a classical problem in combinatorial geometry and raises the best-known exponent for a large distinct-distance subset. The absence of free parameters fitted inside the paper and the reliance on an external theorem plus standard incidence facts are strengths.

major comments (2)
  1. [Proof of main theorem] §3 (or the section containing the proof of the main theorem): the claim that incidence-geometry tools deliver the precise edge-distribution hypotheses required by Li–Postle is load-bearing. The manuscript must exhibit an explicit calculation showing that every subset of vertices spans at most the permitted number of hyperedges (with no logarithmic factors that would reduce the exponent below 1/3).
  2. [Definition of the hypergraph and incidence bounds] The reduction from the geometric instance to the hypergraph H assumes that the multiplicity of each distance is controlled uniformly; if there exist point configurations (e.g., lattices or regular polygons) that produce locally dense subhypergraphs, the Li–Postle bound may fail to apply directly. A separate lemma or remark addressing these extremal configurations is required.
minor comments (2)
  1. [Statement of Theorem 1.1] The implicit constant in |P'| ≫ |P|^{1/3} should be stated explicitly or at least bounded in terms of absolute constants.
  2. [Section 2] Notation for the hypergraph H (vertex set, edge set, uniformity) should be introduced once and used consistently; currently the definition appears piecemeal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the two major comments point by point below. Both concerns can be resolved by adding explicit calculations and a clarifying remark, which we will incorporate in the revised version.

read point-by-point responses

  1. Referee: [Proof of main theorem] §3 (or the section containing the proof of the main theorem): the claim that incidence-geometry tools deliver the precise edge-distribution hypotheses required by Li–Postle is load-bearing. The manuscript must exhibit an explicit calculation showing that every subset of vertices spans at most the permitted number of hyperedges (with no logarithmic factors that would reduce the exponent below 1/3).

    Authors: We agree that the edge-distribution verification is central and benefits from an explicit write-up. The current proof in §3 applies the Szemerédi–Trotter theorem to bound the number of hyperedges spanned by an arbitrary vertex subset S, yielding at most O(|S|^{4/3}) hyperedges (up to constants), which is precisely the threshold required by the Li–Postle theorem. In the revision we will expand this into a self-contained calculation that tracks all constants and confirms the absence of logarithmic factors, thereby preserving the |P'|^{1/3} exponent. revision: yes

  2. Referee: [Definition of the hypergraph and incidence bounds] The reduction from the geometric instance to the hypergraph H assumes that the multiplicity of each distance is controlled uniformly; if there exist point configurations (e.g., lattices or regular polygons) that produce locally dense subhypergraphs, the Li–Postle bound may fail to apply directly. A separate lemma or remark addressing these extremal configurations is required.

    Authors: The incidence-geometry bounds used to define and analyze H are global and apply uniformly; the Szemerédi–Trotter theorem already limits the number of repeated distances in any point set, including lattices and regular polygons. Nevertheless, to make this explicit we will add a short remark immediately after the definition of H that checks the edge-distribution condition on these families and confirms that the same incidence estimates continue to hold, so the Li–Postle hypothesis is satisfied without additional assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies external Li–Postle theorem after verifying distribution conditions via incidence geometry.

full rationale

The paper reduces the problem to finding a large independent set in a distance-repetition hypergraph and invokes the Li–Postle theorem (external, non-overlapping authors) once the required edge-distribution hypotheses are established using standard incidence-geometry bounds. No quantity is defined in terms of itself, no parameter is fitted inside the paper and then renamed a prediction, and the central bound does not reduce to a self-citation chain. The incidence-geometry verification step is independent of the target |P|^{1/3} exponent and does not rely on prior work by the present authors.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of the Li–Postle hypergraph theorem to the distance-repetition hypergraph and on standard incidence bounds; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption The hypergraph whose edges correspond to pairs of pairs sharing a distance satisfies the edge-distribution conditions required by the Li–Postle theorem.
    The paper states that it makes heavy use of this result.

pith-pipeline@v0.9.1-grok · 5632 in / 1074 out tokens · 42365 ms · 2026-06-28T00:52:34.878666+00:00 · methodology

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A combinatorial large sieve for Sidon sets, distances, and norm forms

    math.NT 2026-06 unverdicted novelty 8.0

    A new combinatorial large sieve produces the first super-polylogarithmic upper bounds of the form N exp(-c log N / log log N) for Sidon sets in squares and no-repeated-distance sets in the grid.

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