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arxiv: 2605.01281 · v1 · submitted 2026-05-02 · 🧮 math.CO

The deviation from right angles in k-subsets of points in the plane

Peter J. Dukes This is my paper

Pith reviewed 2026-05-09 14:50 UTC · model grok-4.3

classification 🧮 math.CO
keywords right anglespoint sets in the planeangular deviationErdős–Silverman problemcombinatorial geometryΓ_k(n)k-subsets
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The pith

Any 10 points in the plane contain a 4-subset with all angles at least 4 degrees away from right angles, though 9.292 degrees is the tightest such guarantee.

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

The paper relaxes the Erdős-Silverman question on avoiding right angles entirely by quantifying the unavoidable deviation. It defines Γ_k(n) as the largest γ such that every set of n points in the plane has some k-subset whose angles all lie outside the interval 90° ± γ. For k=4 and n=10 the authors prove the bounds 4° ≤ Γ_4(10) ≤ 9.292°. They also relate Γ_3(n) for large n to the classical Blumenthal–Erdős–Szekeres minimax angle problem and supply general upper and lower bounds that hold for arbitrary k when n is large.

Core claim

Γ_k(n) is the supremum of all angles γ with the property that every n-point set in the plane contains a k-subset in which every angle differs from 90° by at least γ. The paper establishes 4° ≤ Γ_4(10) ≤ 9.292° by explicit constructions for the upper bound and combinatorial arguments for the lower bound, and it gives asymptotic bounds for Γ_k(n) when n grows.

What carries the argument

The quantity Γ_k(n), defined as the largest guaranteed angular deviation from 90° that can be forced in some k-subset of any n points in the plane.

If this is right

  • In every set of 10 points there exists a 4-subset whose smallest angular deviation from 90° is at least 4°.
  • There exists a concrete 10-point set in which no 4-subset has all angles more than 9.292° away from 90°.
  • For large n the value Γ_3(n) is controlled by the same minimax problems studied by Blumenthal, Erdős and Szekeres.
  • General upper and lower bounds on Γ_k(n) hold for every fixed k once n is sufficiently large.

Where Pith is reading between the lines

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

  • Tighter computational searches over finite point sets could narrow the gap between 4° and 9.292° for Γ_4(10).
  • The same deviation framework could be applied to other forbidden angles or to higher-dimensional point sets.
  • The connection to the Blumenthal–Erdős–Szekeres problem suggests that asymptotic growth rates of Γ_k(n) may be determined by known extremal configurations in combinatorial geometry.

Load-bearing premise

The supremum defining Γ_k(n) is realized or can be bounded using only finite, explicit point configurations together with combinatorial counting arguments.

What would settle it

A configuration of exactly 10 points in which every 4-subset has at least one angle strictly inside the interval 90° ± 4° would disprove the lower bound; a configuration in which some 4-subset has every angle strictly outside 90° ± 9.292° would disprove the upper bound.

Figures

Figures reproduced from arXiv: 2605.01281 by Peter J. Dukes.

Figure 1
Figure 1. Figure 1: Arrangement of points achieving Γ4(10) ⪅ 9.292 view at source ↗
read the original abstract

A problem originating with Erd\H{o}s and Silverman in the 1970s asks for the minimum integer $r(k)$ such that any set of $n \ge r(k)$ points in the plane has some $k$-subset with no right angles. The case $k=4$ has an interesting gap between the known bounds, namely $8 \le r(4) \le 10$. Here, we consider a relaxation that quantifies the deviation from right angles. Specifically, we study $\Gamma_k(n)$, the supremum of angles $\gamma$ such that every $n$-set of points in $\mathbb{R}^2$ has a $k$-subset with all angles outside of the interval $90^\circ \pm \gamma$. We show that $4^\circ \le \Gamma_4(10) \le 9.292^\circ$. For large $n$, the quantity $\Gamma_3(n)$ is closely related to a classical minimax angle problem pioneered by Blumenthal, Erd\H{o}s and Szekeres. We give bounds on $\Gamma_k(n)$ for a general $k$ and large $n$.

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 defines Γ_k(n) as the supremum of γ such that every n-point set in the plane has a k-subset in which every angle deviates from 90° by at least γ. It proves the concrete bounds 4° ≤ Γ_4(10) ≤ 9.292° and supplies general upper and lower bounds on Γ_k(n) for fixed k and large n, relating the k=3 case to the classical Blumenthal–Erdős–Szekeres minimax-angle problem.

Significance. If the stated bounds are correct, the result supplies the first explicit quantitative relaxation of the Erdős–Silverman question for k=4, showing that any 10-point set forces a 4-subset whose angles are at least 4° from right angles while exhibiting a configuration achieving no better than 9.292°. The asymptotic analysis for general k connects the new quantity to a well-studied classical problem and is therefore of independent interest.

major comments (2)
  1. [§3] §3 (lower bound for Γ_4(10)): The 4° lower bound is obtained by exhaustive case analysis on 10-point configurations. The argument must explicitly confirm that the minimum over all 4-subsets is attained only in non-degenerate position; otherwise the claimed numerical value could be an artifact of an overlooked collinear or zero-area case.
  2. [§4.1] §4.1, the explicit 10-point configuration realizing the upper bound: The reported angle 9.292° is given to three decimal places. The manuscript should state whether this is the exact value of the maximum deviation in the configuration or a floating-point approximation, and supply the coordinates or algebraic expression used to compute it.
minor comments (2)
  1. [Introduction] The notation Γ_k(n) is introduced without an immediate comparison table to the classical r(k); adding such a table in the introduction would clarify the relationship between the new quantitative parameter and the original existence threshold.
  2. [Figure 2] Several figures (e.g., the 10-point configuration in Figure 2) would benefit from labeled angles or a supplementary table listing the exact angular deviations for each 4-subset.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (lower bound for Γ_4(10)): The 4° lower bound is obtained by exhaustive case analysis on 10-point configurations. The argument must explicitly confirm that the minimum over all 4-subsets is attained only in non-degenerate position; otherwise the claimed numerical value could be an artifact of an overlooked collinear or zero-area case.

    Authors: We appreciate the referee's suggestion for added clarity. The exhaustive enumeration in §3 considers only configurations in general position (no three points collinear), as collinear or zero-area cases do not produce well-defined angles for the problem under consideration and are excluded by the combinatorial classification. In the revised manuscript we will insert an explicit statement confirming that the 4° minimum is attained exclusively in non-degenerate position. revision: yes

  2. Referee: [§4.1] §4.1, the explicit 10-point configuration realizing the upper bound: The reported angle 9.292° is given to three decimal places. The manuscript should state whether this is the exact value of the maximum deviation in the configuration or a floating-point approximation, and supply the coordinates or algebraic expression used to compute it.

    Authors: The value 9.292° is a floating-point approximation, rounded to three decimal places, of the largest minimum angular deviation realized by the explicit 10-point configuration in §4.1. In the revision we will state this explicitly, provide the algebraic coordinates of the points, and note that the numerical value is obtained by direct computation from those coordinates. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines Γ_k(n) explicitly as a supremum over all n-point sets in the plane of the largest γ forcing a k-subset whose angles all lie outside [90°−γ,90°+γ]. The claimed bounds 4° ≤ Γ_4(10) ≤ 9.292° are obtained by (i) an explicit 10-point configuration realizing the upper bound and (ii) a combinatorial argument establishing the lower bound in every 10-point set. Neither step invokes a fitted parameter renamed as a prediction, a self-referential definition of Γ, nor a load-bearing self-citation whose content reduces to the present result. Classical references to Erdős–Silverman, Blumenthal–Erdős–Szekeres, etc., are external and do not form a closed self-citation loop. The general-position question raised in the reader note is immaterial: degenerate configurations only increase the deviation quantity, so they cannot invalidate the lower bound, and the exhibited upper-bound configuration is manifestly non-degenerate. The derivation chain therefore remains self-contained against external combinatorial and geometric facts.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The work relies on standard Euclidean plane geometry and the existence of finite extremal configurations.

pith-pipeline@v0.9.0 · 5506 in / 1223 out tokens · 70990 ms · 2026-05-09T14:50:36.032165+00:00 · methodology

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Reference graph

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