On sets of $n$ points in general position that determine lines that can be pierced by $n$ points

Chaya Keller; Rom Pinchasi

REVIEW 2 major objections 4 minor 19 references

On sets of $n$ points in general position that determine lines that can be pierced by $n$ points

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Under the outside-segment piercing condition, any configuration P∪R of 2n points must lie on a single cubic curve.

desk verdict A genuinely new structural result with an elegant two-step proof, but two real small-case gaps (k=4 hulls and n=5,6) need patching before the theorem as stated is proved. read the letter →

arxiv 1908.06390 v1 pith:77FVBG2H submitted 2019-08-18 math.CO cs.CG

Chaya Keller , Rom Pinchasi This is my paper · ORCID

classification math.COcs.CG MSC 52C1014H50

keywords Erdős–Purdyproblemlinepiercingoutside-segmentconditiongeneralpositioncubiccurveChaslestheoremconvexslopes

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a structural version of the classical line-piercing problem. If $P$ is a set of $n$ points with no three collinear and $R$ is a second set of $n$ points such that every line through two points of $P$ contains a point of $R$ outside the segment between them, then all $2n$ points lie on one cubic curve. The outside-the-segment condition is exactly what makes the configuration rigid: $P$ must be in convex position, every point of $R$ lies outside the hull, and the only collinear triples are those obeying the cyclic rule $i+j+k\equiv 0\pmod n$. Once that structure is known, a Chasles-theorem argument forces the whole set onto a cubic. This confirms the paper's Conjecture 1.2 under a stronger hypothesis and generalizes the earlier extremal characterization of sets with exactly $n$ directions.

What carries the argument

The machinery is two-step. First, the outside-segment hypothesis becomes a rigid combinatorial structure: the points of $P$ are in convex position, the points of $R$ lie outside their convex hull, and collinearities are exactly the modular rule $i+j+k\equiv 0\pmod n$. Second, this rule feeds a Chasles-theorem grid: whenever three lines $\ell_1,\ell_2,\ell_3$ meet three lines $m_1,m_2,m_3$ in nine distinct points, a cubic through any eight of those points passes through the ninth. The paper applies this repeatedly to the nine-point grids formed by the cyclic labels, so that one cubic containing seven $P$-points and two $R$-points must contain every point of $P\cup R$.

What would settle it

A direct way to test the theorem is to search oriented matroids for small $n$, especially hull size $k=4$: if any configuration satisfying the outside-segment condition has $x_i,x_j,r_k$ collinear with $i+j+k\not\equiv 0\pmod n$, or has all $2n$ points not lying on a cubic, the claim is false.

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Extended reading notes

Core claim

The central claim, Theorem 1.3, is that the outside-segment piercing condition is strong enough to determine the entire combinatorial geometry of $P\cup R$ and then force it onto an algebraic curve of degree at most three. More precisely, for every pair $x,y\in P$ the required piercing point $r\in R$ lies outside the interval $xy$; the paper shows $P$ is in convex position, $R$ lies outside $\operatorname{conv}(P)$, and after labelling $P$ cyclically as $x_0,\dots,x_{n-1}$ and $R$ as $r_0,\dots,r_{n-1}$, the collinearity condition is $x_i,x_j,r_k$ collinear iff $i+j+k\equiv 0\pmod n$. It then proves, using Chasles' theorem in the form that a cubic through eight of the nine intersections of two triples of lines passes through the ninth, that any cubic through seven consecutive $P$-points and two $R$-points must contain all of $P\cup R$. Hence $P\cup R$ is contained in a cubic curve.

Load-bearing premise

The load-bearing premise is that for every pair $x,y\in P$ the required piercing point in $R$ lies strictly outside the open segment between $x$ and $y$; without this, the convex-position and cyclic-collinearity rigidity that feeds the Chasles argument has no basis.

Editorial extensions

If this is right

Under the outside-segment hypothesis the extremal piercing configuration is completely rigid: $P$ is in convex position, each point of $R$ lies outside the convex hull, and the only collinear triples are those with $i+j+k\equiv 0\pmod n$.
Consequently every such configuration is algebraic of degree at most three: there is a single cubic curve containing all $2n$ points, so the combinatorial extremum cannot be realized by generic point sets.
The theorem recovers the known characterization of general-position sets with exactly $n$ directions, because the required $R$ can be placed on the line at infinity, outside every finite segment.
The same Chasles propagation works in the bipartite setting: if blue and green points alternate on the convex hull and every blue-green line is pierced by a red point outside the segment, then the union lies on a cubic.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next test is whether the outside-segment condition can be weakened to allow at most one piercing point inside the segment per line; if the cubic conclusion persists, the full Conjecture 1.2 would be much closer.
The cyclic rule $i+j+k\equiv 0\pmod n$ suggests that equality configurations are essentially residues on a cubic curve, so a plausible classification is that they are affine images of regular polygons on a cubic.
A small-case search should target hull sizes $k=3$ and $k=4$, since the counting step invokes a lower bound stated only for even $n>4$; if a counterexample exists, it is most likely there.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear [colleague],

Quick take on Keller–Pinchasi, arXiv:1908.06390. The headline is a genuinely new structural result: under the outside-segment piercing hypothesis, P∪R must lie on a cubic. The two-step proof—first forcing P into convex position with a rigid i+j+k=0 collinearity rule, then using Chasles to get the cubic—is elegant and mostly checkable. I think the paper deserves a real referee, but not as-is: there are two small-case gaps that need to be closed before the theorem as stated is proved.

What is actually new: Theorem 1.3 and the bipartite Theorem 3.1. These generalize Jamison's few-slopes classification to a wider family, and they are the first positive evidence for Conjecture 1.2 under a plausible extra assumption. The Chasles argument in Claims 2.2 and 2.3 is transparent; the reader can verify the nine points are the claimed ones. The combinatorial step—that the outside-segment condition forces convex position and the modular collinearity rule—is the real contribution, and it is not in the cited literature.

Soft spots, in order of seriousness:

1. Small hulls. The proof of |R'|≥k invokes Theorem 1.1, which is stated for even k>4. For k=4 the trivial counting bound only gives |R'|≥3, so the conclusion |R'|=k is not justified. This gap affects any configuration whose convex hull has exactly four vertices, regardless of n.

2. Small n in the Chasles step. The cubic Γ is chosen through x_{n-1},...,x_{n-7}. For n=5 or 6 these indices are not distinct, so Claim 2.2 cannot be applied. No separate argument is given for n=5,6. Thus the theorem as stated is not proved for those n.

Neither gap looks fatal—small cases often get fixed by direct arguments—but the paper as written overclaims. The n≥7, k≥5 (or k≥6) case appears sound to me. The citation pattern is honest; Theorem 1.1 is cited to multiple sources including two manuscripts, which is a bit soft but not unusual for a recent-result-dependent proof. No circularity.

Who this is for: combinatorial geometers working on Erdős–Purdy type problems and on algebraic structure forced by incidences. They will want to read it despite the gaps. Recommendation: send to peer review with a note asking for small-case handling; do not desk reject. If the authors patch the gaps, this is a solid contribution.

Referee Report

2 major / 4 minor

Summary. The paper studies the following problem: given a set P of n points in general position in the plane and a disjoint set R of n points such that every line through two points of P contains a point of R outside the segment determined by those two points, show that P∪R lies on a cubic curve. The main theorem (Theorem 1.3) is proved in two steps. The first step is combinatorial: the outside-segment condition forces P to be in convex position, all points of R to lie outside the convex hull of P, and the collinear triples to follow the explicit rule i+j+k=0 modulo n after suitable labeling. The second step is algebraic: using Chasles's theorem in the style of Green and Tao, the authors construct a cubic curve through seven consecutive points of P and two points of R and then extend containment to the whole set P∪R via two claims (Claims 2.2 and 2.3). A bipartite analogue (Theorem 3.1) is also stated and proved along similar lines.

Significance. If correct, this is a strong and elegant result. It proves a special case of Conjecture 1.2, the conjecture attributed to Erdos and Purdy that equal-size piercing sets force a cubic curve, and it generalizes Jamison's characterization of extremal slope configurations. The proof is largely explicit: the combinatorial classification is clear and the Chasles applications are checkable, following the Green-Tao ordinary-lines framework. The outside-segment hypothesis is the key new input and it is used in a principled way to derive a rigid collinearity structure. No circularity or free parameters appear; external theorems are used as black boxes. The main caveat is that the proof as written has genuine small-case gaps that are not acknowledged in the manuscript.

major comments (2)

[Section 2, paragraph after the definition of P′] The claim that |R′| ≥ k is justified by an application of Theorem 1.1 to the extreme subset P′. However, Theorem 1.1 is stated only for even cardinalities larger than 4. Thus the argument does not cover the case where P′ has exactly four extreme vertices. For odd k the trivial counting bound |R′| ≥ k is indeed available, but this is not stated, and for k = 4 the trivial bound gives only |R′| ≥ 3, which is insufficient. The equality |R′| = k and all subsequent steps depend on this lower bound, so the case of a convex hull with exactly four extreme vertices is not handled. A separate argument for k = 4 (or a modification of the counting argument) is needed for the theorem to hold for all n.
[Section 2.1, construction of the cubic Γ and Claims 2.2–2.3] The algebraic step requires a cubic Γ passing through x_{n−1}, …, x_{n−7} and then repeatedly applies Claim 2.2, whose statement presupposes seven distinct consecutive points of P for the initial configuration and six distinct consecutive points plus three distinct points of R for the Chasles grid. For n = 5 and n = 6 these indices are not distinct modulo n (for example, with n = 6 the list x_5, ..., x_{−1} repeats x_5), so the claims are not applicable. The paper does not provide a separate treatment of n = 5 and n = 6, and the same omission affects the even-n case in Claim 2.3 for n = 6. Thus the main theorem is not proved for n = 5 and n = 6. The same issue affects Theorem 3.1 when 2n is small.

minor comments (4)

[Theorem 1.1 statement] The statement contains a grammatical typo: “Let P is a set” should read “Let P be a set”.
[Throughout the proof] A few typographical errors are present, including “colllinear” in the proof of the collinearity structure and “the the line” in the proof of Claim 2.3. These are harmless but should be corrected.
[Section 2, notational clarity] The term “relevant” is introduced informally. A short formal definition (e.g., “r is relevant for x if there exists y ∈ P \ {x} such that r was chosen for the pair (x,y)”) would improve readability, though the intended meaning is clear from context.
[Figure 4] In the illustration for Claim 2.3, the label “m3” appears to be repeated in a way that may confuse the reader; the intended three m-lines should be labeled distinctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1.3 is derived from the outside-segment hypothesis using external theorems, with no fitted inputs or self-citation chain forcing the conclusion.

full rationale

The paper's central derivation does not reduce to its inputs. The outside-segment condition is used to prove that P is in convex position and that collinearities obey i+j+k=0 modulo n; this structural step is a genuine consequence of the hypothesis, not an assumption dressed as a result. The algebraic step then uses Chasles' theorem (an external geometric fact) to propagate containment in a cubic curve from nine initial points to all of P∪R. No parameter is fitted to data, no quantity called a prediction is defined in terms of the target, and no normalization or ansatz is imported through citation. The cited Theorem 1.1 is used as a black-box lower bound and has independent peer-reviewed proofs ([ABK+08], [Mil18]); the manuscripts [Pin18] and [PP19] merely provide alternative proofs and are not load-bearing for the present argument. The skeptical concerns about n=5,6 and k=4 are proof gaps concerning edge cases, not circularity: they do not show that the conclusion is equivalent to the hypothesis. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper is a pure mathematical proof with no free parameters fitted to data and no new postulated entities. All axioms are standard results in discrete geometry and algebraic geometry. The conclusion is not normalized or fit into existence; the Chasles mechanism is a genuine derivation.

assumptions (4)

standard math Theorem 1.1: for n>4 even, any R piercing all lines determined by n general-position points has |R| at least n.
Used in Section 2 to lower-bound the number of piercing points of the extreme subset P'.
standard math Chasles theorem: if two triples of lines form nine distinct intersections, any cubic through eight of them passes through the ninth.
The engine of the algebraic step, used in Claims 2.2 and 2.3 to propagate containment in a cubic.
standard math Any nine points in the plane lie on some cubic curve.
Used to choose the initial cubic Gamma through nine specified points in the second part of Section 2.
domain assumption General position: no three points of P or of B union G are collinear.
Stated in the theorems and used throughout the counting and collinearity arguments.

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Pith. "Pith review of On sets of $n$ points in general position that determine lines that can be pierced by $n$ points." pith.science (2026). https://pith.science/paper/77FVBG2H

@misc{pith2026190806390,
  author       = {Pith},
  title        = {Pith review of: On sets of $n$ points in general position that determine lines that can be pierced by $n$ points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/77FVBG2H}},
  note         = {Machine review of arXiv:1908.06390}
}

abstract

Let $P$ be a set of $n$ points in general position in the plane. Let $R$ be a set of $n$ points disjoint from $P$ such that for every $x,y \in P$ the line through $x$ and $y$ contains a point in $R$ outside of the segment delimited by $x$ and $y$. We show that $P \cup R$ must be contained in a cubic curve.

Figures

Figures reproduced from arXiv: 1908.06390 by the authors.

**Figure 1.** Constructions with |R| = n − 1 for n = 2, 4. The points in P are colored black while the points in R are colored white. In [EP78] Erd˝os and Purdy considered the same problem in the case where the set P is in general position in the sense that no three points of P are collinear. In this case if n is odd the ∗Mathematics Department, Technion – Israel Institute of Technology, Haifa 32000, Israel. chayak@technion.ac.il… view at source ↗

**Figure 2.** Illustration for the proof that no point of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Illustration for the proof of Claim 2.2 with [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Illustration for the proof of Claim 2.3 with [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]

**Figure 5.** Figure 5: Illustration for the proof of Theorem 3.1. [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

[1]

Ackerman, K

E. Ackerman, K. Buchin, C. Knauer, R. Pinchasi, and G. Rote. There are not too many magic configurations. Discrete & Computational Geometry , 39(1):3--16, 2008

work page 2008
[2]

Blokhuis, G

A. Blokhuis, G. Marino, and F. Mazzocca. Generalized hyperfocused arcs in PG (2,p). Journal of Combinatorial Designs , 22(12):506--513, 2014

work page 2014
[3]

Brass, W

P. Brass, W. O. J. Moser, and J. Pach. Research Problems in Discrete Geometry . Springer US, 2005

work page 2005
[4]

G. D. Chakerian. Sylvester's problem on collinear points and a relative. Amer. Math. Monthly , 77:164--167, 1970

work page 1970
[5]

G. Elekes. On linear combinatorics III . F ew directions and distorted lattices. Combinatorica , 19(1):43--53, 1999

work page 1999
[6]

Erd o s and G

P. Erd o s and G. Purdy. Some combinatorial problems in the plane. Journal of Combinatorial Theory, Series A , 25(2):205--210, 1978

work page 1978
[7]

Gr \" u nbaum

B. Gr \" u nbaum. Arrangements of colored lines, abstract 720-50-5. Notices Amer. Math. Soc. , 22:A--200, 1975

work page 1975
[8]

Green and T

B. Green and T. Tao. On sets defining few ordinary lines. Discrete & Computational Geometry , 50(2):409--468, 2013

work page 2013

Show all 19 references

[9]

R. E. Jamison. A survey of the slope problem. Ann. New York Acad. Sci. , 440(1):34--51, 1985

1985
[10]

R. E. Jamison. Few slopes without collinearity. Discrete Mathematics , 60:199--206, 1986

1986
[11]

Mili \' c evi \' c

L. Mili \' c evi \' c . Classification theorem for strong triangle blocking arrangements. arXiv:1809.08639, 2018

2018 arXiv
[12]

T. S. Motzkin. Nonmixed connecting lines, abstract 67 T 605. Notices Amer. Math. Soc. , 14:837, 1967

1967
[13]

U. S. R. Murty. How many magic configurations are there? Amer. Math. Monthly , 78(9):1000--1002, 1971

1971
[14]

C. Pilatte. On the sets of n points forming n+1 directions. arXiv:1811.01055, 2018

2018 arXiv
[15]

Pinchasi

R. Pinchasi. A solution to a problem of G r \"u nbaum and M otzkin and of E rd o s and P urdy about bichromatic configurations of points in the plane. Israel Journal of Mathematics , 198(1):205--214, 2013

2013
[16]

Pinchasi

R. Pinchasi. Stabbing all lines determined by points in general position in the plane. Manuscript, 2018

2018
[17]

Pinchasi and A

R. Pinchasi and A. Polyanskii. A one-page solution of a problem of E rd o s and P urdy. Manuscript, 2019

2019
[18]

P. R. Scott. On the sets of directions determined by n points. Amer. Math. Monthly , 77(5):502--505, 1970

1970
[19]

P. Ungar. 2n noncollinear points determine at least 2n directions. J. Combin. Th. Ser. A , 33(3):343--347, 1982

1982

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Ackerman, K

E. Ackerman, K. Buchin, C. Knauer, R. Pinchasi, and G. Rote. There are not too many magic configurations. Discrete & Computational Geometry , 39(1):3--16, 2008

work page 2008

[2] [2]

Blokhuis, G

A. Blokhuis, G. Marino, and F. Mazzocca. Generalized hyperfocused arcs in PG (2,p). Journal of Combinatorial Designs , 22(12):506--513, 2014

work page 2014

[3] [3]

Brass, W

P. Brass, W. O. J. Moser, and J. Pach. Research Problems in Discrete Geometry . Springer US, 2005

work page 2005

[4] [4]

G. D. Chakerian. Sylvester's problem on collinear points and a relative. Amer. Math. Monthly , 77:164--167, 1970

work page 1970

[5] [5]

G. Elekes. On linear combinatorics III . F ew directions and distorted lattices. Combinatorica , 19(1):43--53, 1999

work page 1999

[6] [6]

Erd o s and G

P. Erd o s and G. Purdy. Some combinatorial problems in the plane. Journal of Combinatorial Theory, Series A , 25(2):205--210, 1978

work page 1978

[7] [7]

Gr \" u nbaum

B. Gr \" u nbaum. Arrangements of colored lines, abstract 720-50-5. Notices Amer. Math. Soc. , 22:A--200, 1975

work page 1975

[8] [8]

Green and T

B. Green and T. Tao. On sets defining few ordinary lines. Discrete & Computational Geometry , 50(2):409--468, 2013

work page 2013