arxiv: 2605.00918 · v1 · submitted 2026-04-30 · 🧮 math.CO · math.MG

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Visibility cliques, cubic containers, and dense orchard cores

Sohail Sarkar

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Pith reviewed 2026-05-09 20:48 UTC · model grok-4.3

classification 🧮 math.CO math.MG

keywords visible cliquescubic curvesplanar point setsBig-Line-Big-Clique conjectureordinary linesorchard problemcollinear points

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The pith

If all but s points of an n-point planar set lie on a cubic curve with no k collinear, the set contains a visible clique of size Omega(n/(s+1)) unless the curve is three lines.

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

The paper proves a cubic-container theorem showing that when most points of a finite planar set lie on a single real cubic, the visibility graph on those points can be covered by O(s) cliques where s counts the points off the curve. This immediately gives a large clique of pairwise visible points whose size is linear in n divided by s, provided no k points are collinear and the cubic is not the degenerate three-line case. The same container is combined with the Green-Tao theorem on ordinary lines and the Elekes-Szabo theorem on triple lines to settle the Big-Line-Big-Clique conjecture for all sets lying on a fixed irreducible algebraic curve and for all sets that have only linearly many ordinary lines. A separate dense-orchard core lemma shows that the total absence of a large visible clique forces a positive-density subset in which every point lies on many 3-rich lines.

Core claim

If A subset R^2 has n points, no k collinear points, and all but s points of A lie on a real cubic, then the cubic-supported part of A has a visible clique cover of size O_k(s+1); in particular V(A) contains a clique of size Omega_k(n/(s+1)), unless the cubic is the excluded three-line case containing only O_k(1) points. Combining this with the Green-Tao structure theorem yields that every n-point set with no k collinear points and at most Kn ordinary lines contains a visible clique of size Omega_{k,K}(n) and can have all but O_K(1) points partitioned into O_{k,K}(1) mutually visible sets. The same container plus the Elekes-Szabo theorem proves the Big-Line-Big-Clique conclusion for any set.

What carries the argument

The cubic-container theorem, which produces an O_k(s+1)-sized visible clique cover for the points of A that lie on the cubic.

If this is right

Every n-point set with no k collinear points and at most Kn ordinary lines contains a visible clique of linear size and can be partitioned into O(1) mutually visible sets after removing O(1) points.
The Big-Line-Big-Clique conjecture holds for every point set contained in a fixed irreducible algebraic curve.
The absence of a visible K_ℓ forces a positive-density subset in which every point lies on linearly many 3-rich lines.
Ambient blockers such as the three-line cubic must be treated separately from the generic cubic case.

Where Pith is reading between the lines

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

General point sets without large visible cliques might still contain large subsets lying on low-degree curves that can be handled by the same container argument.
The dense-orchard core lemma could be iterated to produce further combinatorial structure when visible cliques are small.
The sharp one-blocker example indicates that any proof of the full conjecture must explicitly control or exclude degenerate ambient configurations.

Load-bearing premise

All but s points lie on one real cubic curve that is not the union of three lines, and the set contains no k collinear points.

What would settle it

An explicit construction of n points with s off a non-degenerate cubic, no k collinear, yet whose visibility graph has maximum clique size o(n/s).

Figures

Figures reproduced from arXiv: 2605.00918 by Sohail Sarkar.

**Figure 1.** Figure 1: The cubic y 2 = x 3 − x in its standard affine chart. The squares mark the three finite vertical tangencies (−1, 0), (0, 0) and (1, 0). The dots mark the two finite flexes, and the point at infinity E∞ = [0 : 1 : 0], indicated by the arrows, is the third real flex. Removing these exceptional points gives the labelled visibility patches. The labels V4, V6 denote the two unbounded affine arcs, which meet at … view at source ↗

read the original abstract

The Big-Line-Big-Clique Conjecture of Kara, Por and Wood asserts that, for every fixed $k$ and $\ell$, every sufficiently large finite planar point set contains either $k$ collinear points or $\ell$ pairwise visible points. We prove a quantitative form in two structured regimes and isolate the precise ambient obstruction to the full conjecture. The main result is a deterministic cubic-container theorem. If $A \subset \mathbb{R}^2$ has $n$ points, no $k$ collinear points, and all but $s$ points of $A$ lie on a real cubic, then the cubic-supported part of $A$ has a visible clique cover of size $O_k(s+1)$; in particular $V(A)$ contains a clique of size $\Omega_k(n/(s+1))$, unless the cubic is the excluded three-line case containing only $O_k(1)$ points. Combining this with the Green-Tao structure theorem, we obtain that every $n$-point set with no $k$ collinear points and at most $Kn$ ordinary lines contains a visible clique of size $\Omega_{k,K}(n)$; more strongly, all but $O_K(1)$ points can be partitioned into $O_{k,K}(1)$ mutually visible sets. We also combine the cubic-container theorem with the Elekes-Szabo theorem on triple lines and cubic curves to prove the Big-Line-Big-Clique conclusion for point sets contained in any fixed irreducible algebraic curve. Finally, we prove a dense-orchard core lemma showing that the absence of a visible $K_\ell$ forces a positive-density subset in which every point lies on linearly many 3-rich lines, and we give a sharp one-blocker example showing why ambient blockers cannot be ignored.

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 cubic-container theorem cleanly resolves quantitative versions of the Big-Line-Big-Clique conjecture for points mostly on cubics and for sets with few ordinary lines.

read the letter

This paper proves a deterministic cubic-container theorem: if an n-point set in the plane has no k collinear points and all but s points lie on a real cubic (not the three-line case), then the cubic-supported subset admits a visible-clique cover of size O_k(s+1). In particular this yields a visible clique of size Omega_k(n/(s+1)). The same container combines with Green-Tao to show that sets with at most K n ordinary lines contain a visible clique of size Omega_{k,K}(n) and can be partitioned into O_{k,K}(1) mutually visible sets after removing O_K(1) points. It also pairs with Elekes-Szabo to settle the conjecture for points on any fixed irreducible algebraic curve. The dense-orchard core lemma is a useful side result showing that the absence of a large visible clique forces a positive-density subset with many 3-rich lines, and the one-blocker example clarifies why ambient structure cannot be ignored.

Referee Report

0 major / 2 minor

Summary. The paper proves a quantitative cubic-container theorem for the Big-Line-Big-Clique Conjecture: if an n-point set A in R^2 has no k collinear points and all but s points lie on a real cubic (excluding the three-line case), then the cubic-supported subset admits a visible-clique cover of size O_k(s+1), hence V(A) contains a clique of size Omega_k(n/(s+1)). Combining this with the Green-Tao structure theorem yields Omega_{k,K}(n)-sized visible cliques (and an O_{k,K}(1)-partition into mutually visible sets) for n-point sets with at most Kn ordinary lines. The same container theorem plus the Elekes-Szabo theorem on triple lines and cubics establishes the conjecture for all point sets contained in a fixed irreducible algebraic curve. The paper also proves a dense-orchard core lemma (absence of K_ell forces a positive-density subset in which every point lies on linearly many 3-rich lines) and supplies a sharp one-blocker example.

Significance. If the derivations hold, the work supplies the first deterministic container theorem in this setting and resolves the conjecture in two natural structured regimes (few ordinary lines; points on a fixed curve). The explicit dependence on s and k, the non-circular use of Green-Tao and Elekes-Szabo, and the clarifying dense-orchard lemma plus blocker example are genuine strengths that advance the field beyond the original conjecture statement.

minor comments (2)

Abstract, line 3: the notation V(A) for the visibility graph is introduced without a forward reference to its definition in §2; a parenthetical pointer would improve readability.
The O_k and Omega_k notation is used throughout; while the k-dependence is expected, a brief remark in the introduction on whether the constants are effective (or at least computable from the cited theorems) would be helpful for applications.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive report, the accurate summary of our results, and the recommendation of minor revision. The assessment correctly identifies the deterministic cubic-container theorem, its combination with Green-Tao and Elekes-Szabo, the dense-orchard core lemma, and the one-blocker example as the main contributions.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives its central cubic-container theorem directly from the no-k-collinear assumption and the geometry of points lying on a real cubic (excluding the three-line case). This new result is then combined with the Green-Tao structure theorem and the Elekes-Szabo theorem on triple lines and cubics, both of which are independently established prior results whose statements and proofs are external to the current paper and do not rely on its claims. No self-definitional reductions, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via prior work by the same author appear in the derivation chain. The overall argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on two major prior theorems plus standard facts from real algebraic geometry; no new free parameters or invented entities are introduced beyond k-dependent constants implicit in the O-notation.

free parameters (1)

k-dependent constants in O_k and Omega_k
The bounds are stated with constants depending on k but not computed explicitly; these are treated as existing rather than fitted.

axioms (2)

standard math Green-Tao structure theorem on sets with few ordinary lines
Invoked to obtain the Omega(n) visible clique when the number of ordinary lines is at most Kn.
standard math Elekes-Szabo theorem on triple lines and cubic curves
Used to extend the cubic-container result to point sets contained in any fixed irreducible algebraic curve.

pith-pipeline@v0.9.0 · 5628 in / 1563 out tokens · 35370 ms · 2026-05-09T20:48:43.348942+00:00 · methodology

discussion (0)

Reference graph

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