Subset Selection Problems in Planar Point Sets

Adrian Dumitrescu; Dingyuan Liu; Felix Christian Clemen; J\'ozsef Balogh

REVIEW 1 major objections 5 minor 34 references

Subset Selection Problems in Planar Point Sets

T0 review · 1 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For planar points with bounded collinearity, the largest no-three-in-line subset is $\Theta(n/s)$ when $s$ is large, with new bounds in every remaining regime.

desk verdict Genuinely new bounds, but the container-step verification has a repairable log-factor gap. read the letter →

arxiv 2412.14287 v1 pith:QL3RLT5X submitted 2024-12-18 math.CO cs.CG

József Balogh , Felix Christian Clemen , Adrian Dumitrescu , Dingyuan Liu This is my paper · ORCID

classification math.COcs.CG MSC 52C1005D4005C65

keywords subsetselectiongeneralpositioncollinearitymonotonesubsetsdistinctslopeshypergraphcontainerssupersaturationplanarpointsets

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 establishes upper bounds for three subset-selection problems on planar point sets with limited collinearity. For the classical problem of choosing a subset with no three points on a line, the main result is that every $n$-point set with at most $s$ points per line has such a subset of size $O(n/s)$ when $n^{1/3} \le s \le n$, and $O(n^{5/6+o(1)}/\sqrt{s})$ when $3 \le s \le n^{1/3}$; together with previously known lower bounds this gives $\Theta(n/s)$ once $s = \Omega(\sqrt{n} \log n)$. The paper also proves bounds for monotone general-position subsets, showing upper $O((n/s)^{1/2})$ and, when $s=\Theta(\sqrt{n})$, upper $O(n^{1/4})$ and lower $\Omega((n \log \log n/\log n)^{1/4})$, so the gap is only logarithmic. For subsets with pairwise distinct slopes it gives a universal lower bound $\Omega((n/\log s)^{1/3})$ and worst-case upper bounds $O(\sqrt{n})$ and $O((n/s)^{4/5})$ in complementary ranges of $s$. These are quantitative answers to questions rooted in classical no-three-in-line problems that remain open for grids.

What carries the argument

The load-bearing object is the 3-uniform hypergraph whose vertices are grid points and whose edges are collinear triples. The new step is a balanced supersaturation lemma: any $S\subseteq[n]^3$ of size $n^{3-x}$ contains a spanning subhypergraph $H'$ with $e(H')\ge n^{6-4x}/(10^7\log n)$, maximum degree at most $n^{3-3x}/(10^3 f\log n)$, and pair-codegree at most 1. Feeding this hypergraph into the container lemma yields containers that shrink by a constant fraction, and iterating $(\log n)^2$ times produces $\exp(n^{(3+\gamma)/2+o(1)})$ containers of size $n^{3-\gamma+f}$. A random subset of $[n]^3$ with no collinear $n^\alpha$-tuple is then shown, by a first-moment estimate over the containers, to have no large general-position subset; projecting back to the plane gives the upper-bound construction.

What would settle it

Compute the edge count in Lemma 11 directly for a small example such as $S=[n]^3$ with $x=0$: because each line contributes a matching of about $|\ell\cap S|/3$ edges rather than $\binom{|\ell\cap S|}{3}$, the displayed lower bound $e(H')\ge n^6/(10^7\log n)$ should be checked against the corrected count; if it falls below the bound, the supersaturation lemma loses support. Separately, a search for $n$-point sets with at most 4 collinear points and a general-position subset larger than the claimed $O(n^{5/6+o(1)}/\sqrt{4})$ bound would refute Theorem 1(ii) in that case.

Watch

Extended reading notes

Core claim

The paper's central claim is Theorem 1: for every $n$-point planar set $P$ with at most $s$ points on any line, the largest general-position subset has size $f(n,s)=O(n/s)$ for $n^{1/3}\le s\le n$ and $f(n,s)=O(n^{5/6+o(1)}/\sqrt{s})$ for $3\le s\le n^{1/3}$; in particular $f(n,s)=\Theta(n/s)$ when $s=\Omega(\sqrt{n}\log n)$. The large-$s$ range is proved by taking a random subset of the square grid $[n]^2$ and applying the pigeonhole principle. The small-$s$ range is proved by lifting to the cube $[n]^3$, applying the hypergraph container method to a sparsified hypergraph of collinear triples with pair-codegree at most one, iterating containers until every collinear-triple-free set lies in a small family of small containers, and then projecting a random subset back to the plane. The paper additionally claims bounds for monotone general-position subsets and for subsets with pairwise distinct slopes, stated as Theorems 4 through 6.

Load-bearing premise

The proof of the small-$s$ range in Theorem 1(ii) relies on equation (33), which derives the factor $1/\sqrt{s}$ for $3\le s\le\log n$ from monotonicity alone; that division is not a consequence of monotonicity, and the container argument only establishes it for $s\ge\log n$.

Editorial extensions

If this is right

If Theorem 1 is right, the general-position problem is resolved up to constants for $n^{1/3}\le s\le n$, and exactly $\Theta(n/s)$ for $s=\Omega(\sqrt{n}\log n)$.
When $s=\Theta(\sqrt{n})$, monotone general-position subsets have size between $\Omega((n\log\log n/\log n)^{1/4})$ and $O(n^{1/4})$, so only logarithmic factors separate the bounds.
Every $\sqrt{n}\times\sqrt{n}$ grid contains a monotone general-position subset of size $\Omega((n/\log n)^{2/5})$, while every such subset has size $o(n^{1/2})$.
For distinct slopes, any $n$-point set with at most $O(\sqrt{n})$ collinear points contains a subset of size $\Omega((n/\log s)^{1/3})$, matching the best grid bound when $s=\Theta(\sqrt{n})$, with worst-case upper bounds $O(\sqrt n)$ for $s\le n^{3/8}$ and $O((n/s)^{4/5})$ after.
As a corollary, any $n$-point set with at most $\sqrt n$ collinear points can be colored with $O(\sqrt n\log n)$ colors so each color class is in general position.

Reading between the lines

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

The proof of the small-$s$ range in Theorem 1(ii) contains a monotonicity step that is not fully justified for $3\le s\le\log n$; if that step fails, the claimed $1/\sqrt{s}$ improvement for fixed small $s$ would be unsupported even though the rest of the theorem could stand.
If the small-$s$ bound survives scrutiny, it predicts a genuine constant-factor improvement over $f(n,3)$ for every fixed $s>3$, which could be probed computationally on small grids for $s=4,5$.
The balanced supersaturation lemma is formulated for collinear triples in $[n]^3$, but the same container iteration only needs bounded pair-codegrees; it may transfer to higher-dimensional grids or to collinear $k$-tuples.
The distinct-slopes bounds leave a wide gap in the middle range, suggesting the extremal construction is neither a grid nor a parabolic Sidon set; finding the true extremal set for $s$ near $n^{3/8}$ is a concrete next step.

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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

Worth engaging with. The main upper-bound proof as written has a gap in the container step, but it is local and easy to fix. The reader's objection to equation (33) does not survive reading: for s ≤ log n, sqrt(s) = n^{o(1)}, so the known f(n,3) bound yields exactly the claimed O(n^{5/6+o(1)}/sqrt(s)) once the o(1) exponent absorbs sqrt(log n). Monotonicity is fine here.

The real issue is equation (36). The second term in Δ(H',τ) is 10^8 log n / n^f, which is not O(n^{-f}) as the stated ε = n^{-f}/2 requires. So Lemma 10 cannot be applied with that ε. The fix is simple: take ε to be a small fixed constant, e.g., 1/4; then the condition holds for sufficiently large n, and the container-size estimate only needs ε bounded away from 1. This should be corrected in revision, but it is not fatal.

What is actually new: the balanced supersaturation lemma (Lemma 11) and its use inside the container iteration is a genuine technical novelty, and it yields the first sub-polynomial improvement over the f(n,3) upper bound for intermediate s. The lower bounds for g and h on general point sets, and the near-matching logarithmic gaps at s = Θ(√n), are also new. The paper is clearly written, and the tools — incidence geometry, containers, additive combinatorics — are used appropriately.

Soft spots beyond (36): Lemma 11 has a typo in the displayed edge-count formula, writing binom(|ℓ∩S|,3) where the matching gives about |ℓ∩S|/3; the subsequent lower bound uses the correct quantity, so it is cosmetic. The small-s part of Theorem 1(ii) is fine once you accept the monotonicity-plus-o(1) argument. The proof of Theorem 4's lower bound is a quick combination of Proposition 2 and Erdős–Szekeres, which is legitimate.

For whom: anyone working on subset selection in discrete geometry or on container-method applications to geometric problems. It deserves a serious referee; the gap is local and repairable. I would send it out.

Referee Report

1 major / 5 minor

Summary. The paper studies three subset-selection problems for n-point planar sets with maximum collinearity s: (i) the largest subset in general position, (ii) the largest monotone general-position subset, and (iii) the largest subset with pairwise distinct slopes. The main results are Theorem 1, giving upper bounds f(n,s)=O(n/s) for n^{1/3} ≤ s ≤ n and f(n,s)=O(n^{5/6+o(1)}/√s) for 3 ≤ s ≤ n^{1/3}; Theorem 4, giving g(n,s)=Ω((n log n log s / s^2 / log s)^{1/4}) and O((n/s)^{1/2}); Theorem 5, giving lower and upper bounds for the grid; Theorem 6, giving h(n,s)=Ω((n/log s)^{1/3}) and upper bounds; and Theorem 22, on coloring point sets into general-position classes. The proofs combine incidence geometry, the Cooper–Mubayi coloring lemma, the Saxton–Thomason container method, Szemerédi–Trotter, Sidon sets, and random sampling.

Significance. If the proofs are correct, the paper provides the first upper bounds for f(n,s) in the non-constant range that are tight up to constants when s=Ω(√n log n), as well as new bounds for the monotone and distinct-slope variants for arbitrary planar sets. The container-based approach extends Balogh–Solymosi in a novel way, and the lower-bound constructions are clean and use a well-chosen mix of existing tools. However, the central container step contains a false inequality in §3.2, Eq. (36), which currently invalidates the proof of Theorem 1(ii); the gap appears easily repairable, so the results are likely salvageable with a modest revision.

major comments (1)

[§3.2, Eq. (36)] The application of Lemma 10 fails as written. With d ≥ n^{3−3x}/(10^7 log n), τ = n^{x+γ/2−3/2}, and x ≤ γ − f, the second term in Δ(H′,τ) satisfies 2Δ_3/(dτ^2) ≤ 2·10^7 log n / n^{γ−x} ≤ 2·10^7 log n / n^f, so Δ(H′,τ) = O(log n / n^f), not O(n^{−f}). For the claimed ε = n^{−f}/2, the required inequality Δ(H′,τ) ≤ ε/(100·3!) = n^{−f}/1200 would force log n = O(1), which is false for large n. Consequently, Lemma 10 cannot be applied with this ε, and the container collection, the shrinking bound (38), and the container-size estimate (39) are not justified as written. A simple repair is to take ε to be a small absolute constant, e.g., ε = 1/4; then Δ(H′,τ) ≤ 1/2400 for sufficiently large n, and the later steps (37)–(39) only require that ε is bounded away from 1. The repair should be stated explicitly because the current text claims ε = n^{−f}/2.

minor comments (5)

[§3.2, Eq. (33)] The monotonicity step in Eq. (33) is actually valid: for 3 ≤ s ≤ log n, we have √s ≤ √(log n) = n^{o(1)}, so the o(1) in the known bound f(n,3) ≤ n^{5/6+o(1)} absorbs the extra √(log n) factor. The claimed bound O(n^{5/6+o(1)}/√s) therefore follows for this range; the concern that monotonicity alone cannot create the √s improvement is unfounded, because the improvement comes from the slack in the exponent, not from a hidden monotonicity inequality.
[Lemma 11, Eq. (31)] The displayed equality e(H'[ℓ∩S]) = binom(|ℓ∩S|,3) is incorrect: H' is defined by placing a matching on ℓ∩S, so the number of edges on a line ℓ is floor((|ℓ∩S|−2)/3), not a binomial coefficient. The subsequent lower bound T_v ≥ |S|/3 − |L_v| is consistent with the matching interpretation, so this is a presentation typo rather than a substantive error, but it should be corrected.
[Lemma 11] The bound on Δ(H′) is written as n^{3−3x} 10^{3f} log n, which is ambiguous: it could be read as n^{3−3x}/(10^{3f} log n) or n^{3−3x}/(10^3 f log n). The proof and the subsequent use of the bound in Eq. (36) depend on the intended denominator; the authors should disambiguate the notation.
[§4.1] In the upper-bound construction for Theorem 4, the phrase 'slightly randomly perturb each of its grid points and replace it by a collinear s-tuple' should specify that the perturbed points are chosen so that no unintended collinearities or monotonicity obstructions arise; as written, the construction is rather informal, though the intended argument is clear.
[§5.1] In the proof of Theorem 6 lower bound, the notation L is used both for the number of collinear triples and, in Lemma 18, for a line; this is not a mathematical error but the double use of L may confuse readers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is self-contained and any self-citations are to independent published results.

full rationale

The paper's central claims are proved by explicit constructions and by standard external theorems, not by assuming the conclusions. Theorem 1(i) uses a random grid construction and Chernoff bounds; Theorem 1(ii) uses the Saxton–Thomason container lemma, a new supersaturation/sparsification lemma, and the published Balogh–Solymosi framework. The self-citations to Balogh–Morris–Samotij [1] and Balogh–Solymosi [2] reference published mathematical results whose stated assumptions do not include the target bounds; they are external evidence rather than circular support. Equation (33) is not circular: monotonicity f(n,s) ≤ f(n,3) plus the known bound f(n,3) ≤ n^{5/6+o(1)} gives the claimed form for 3 ≤ s ≤ log n because √s ≤ √(log n) = n^{o(1)} is absorbed by the o(1) exponent. No fitted parameter is renamed as a prediction, no definitional identity forces a target quantity, and no uniqueness claim is imported from the authors' prior work to forbid alternatives. The possible technical issue raised about equation (36) concerns whether the container lemma's hypotheses are verified; that is a correctness or gap concern, not a circularity, since it does not reduce the derived bound to an input by construction. The distinct-slope upper bounds similarly rest on known Sidon-set facts and grid constructions, not on the theorem being proved. Overall, the derivation is self-contained against external benchmarks, and the honest finding is no circularity.

Assumptions & free parameters 0 free parameters · 10 assumptions · 0 invented entities

The proofs are purely mathematical and introduce no fitted numerical parameters, no ad hoc physical or geometric entities, and no claimed empirical inputs. All background results are stated standard theorems from the cited literature.

assumptions (10)

standard math Szemerédi-Trotter theorem and its line-count corollary
Used in Lemma 8 and Lemma 18 to bound the number of collinear triples and trapezoids.
standard math Saxton-Thomason hypergraph container theorem
The central tool in the proof of Theorem 1(ii).
standard math Cooper-Mubayi sparse hypergraph coloring theorem
Used in Proposition 2 to lower bound the independence number of the collinear-triple hypergraph.
standard math Chernoff bounds
Used in the random grid constructions in Theorem 1, Theorem 5, and Theorem 6.
standard math Erdős-Szekeres monotone subsequence theorem
Used in Theorem 4 to extract a monotone subset from a general-position subset.
standard math Prime number estimates of Rosser-Schoenfeld
Used in Lemma 11 to count prime directions in the line family.
standard math Jarník's bound on lattice points on convex curves
Cited in the context of the Theorem 5 lower bound; the proof uses a random sector of an annulus instead.
standard math Pomerance's collinear subset theorem for lattice point sequences
Used in the Theorem 5 upper bound to force a collinear triple in any large monotone grid subset.
standard math Sidon set maximum size at most 2*sqrt(n)+1
Used in Theorem 6 upper bound (i) to show every 2*sqrt(n)+2 points on the parabola contain a trapezoid.
standard math Gauss circle problem estimate
Used in Theorem 5 to count grid points in the annulus sector.

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Pith. "Pith review of Subset Selection Problems in Planar Point Sets." pith.science (2026). https://pith.science/paper/QL3RLT5X

@misc{pith2026241214287,
  author       = {Pith},
  title        = {Pith review of: Subset Selection Problems in Planar Point Sets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QL3RLT5X}},
  note         = {Machine review of arXiv:2412.14287}
}

abstract

Given a finite set satisfying condition $\mathcal{A}$, the subset selection problem asks, how large of a subset satisfying condition $\mathcal{B}$ can we find? We make progress on three instances of subset selection problems in planar point sets. Let $n,s\in\mathbb{N}$ with $n\geq s$, and let $P\subseteq\mathbb{R}^2$ be a set of $n$ points, where at most $s$ points lie on the same line. Firstly, we select a general position subset of $P$, i.e., a subset containing no $3$ points on the same line. This problem was proposed by Erd\H{o}s under the regime when $s$ is a constant. For $s$ being non-constant, we give new lower and upper bounds on the maximum size of such a subset. In particular, we show that in the worst case such a set can have size at most $O(n/s)$ when $n^{1/3}\leq s\leq n$ and $O(n^{5/6+o(1)}/\sqrt{s})$ when $3\leq s\leq n^{1/3}$. Secondly, we select a monotone general position subset of $P$, that is, a subset in general position where the points are ordered from left to right and their $y$-coordinates are either non-decreasing or non-increasing. We present bounds on the maximum size of such a subset. In particular, when $s=\Theta(\sqrt{n})$, our upper and lower bounds differ only by a logarithmic factor. Lastly, we select a subset of $P$ with pairwise distinct slopes. This problem was initially studied by Erd\H{o}s, Graham, Ruzsa, and Taylor on the grid. We show that for $s=O(\sqrt{n})$ such a subset of size $\Omega((n/\log{s})^{1/3})$ can always be found in $P$. When $s=\Theta(\sqrt{n})$, this matches a lower bound given by Zhang on the grid. As for the upper bound, we show that in the worst case such a subset has size at most $O(\sqrt{n})$ for $2\leq s\leq n^{3/8}$ and $O((n/s)^{4/5})$ for $n^{3/8}\leq s=O(\sqrt{n})$. The proofs use a wide range of tools such as incidence geometry, probabilistic methods, the hypergraph container method, and additive combinatorics.

Figures

Figures reproduced from arXiv: 2412.14287 by the authors.

**Figure 1.** Left: Dudeney’s maximum-sized general position subset in the 8 × 8 grid. Right: a monotone general position subset in the 8 × 8 grid. 1.3. Selecting a subset with pairwise distinct slopes. A strengthening of the general position requirement is that of pairwise distinct slopes. Indeed every point set with pairwise distinct slopes has no collinear triple. For the √ n × √ n grid G, Erd˝os, Graham, Ruzsa, and Taylor [11… view at source ↗

**Figure 2.** Upper bound construction for Theorem 4 when n = 36 and s = 4. In this construction there are n points with at most s collinear points. Let P denote the resulting set and let A ⊆ P be a monotone general position subset. Note that (i) A contains at most two points from each collinear s-tuple; (ii) the largest monotone subset of the k × k grid that we started with has at most 2k − 1 points. Thus, |A| ≤ 4k − 2 = O((n/s)… view at source ↗

**Figure 3.** Part (a) illustrates the two parallel lines ℓ, ℓ′ from the proof of Lemma 13. Part (b) illustrates the points a, b, c on the line ℓ from the proof of Claim 15. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: for an illustration. Note that |B ∩ G| = Θ(|A ∩ G|) = Θ(x √ n). v1 v2 v3 o [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

J´ ozsef Balogh, Robert Morris, and Wojciech Samotij, Independent sets in hypergraphs,Journal of the American Mathematical Society 28(3) (2015), 669–709

work page 2015

[2] [2]

J´ ozsef Balogh and J´ ozsef Solymosi, On the number of points in general position in the plane,Discrete Analysis 16 (2018), 1–20

work page 2018

[3] [3]

Imre B´ ar´ any and Maria Prodromou, On maximal convex lattice polygons inscribed in a plane convex set,Israel Journal of Mathematics 154(1) (2006), 337–360

work page 2006

[4] [4]

Peter Braß, William Moser, and J´ anos Pach, Research Problems in Discrete Geometry , Springer, New York, 2005

work page 2005

[5] [5]

Yaobin Chen, Xizhi Liu, Jiaxi Nie, and Ji Zeng, Random Tur´ an and counting results for general position sets over finite fields, Preprint arXiv:2309.07744, 2024

work page Pith review arXiv 2024

[6] [6]

Felix Christian Clemen, Applications of sparse hypergraph colorings, Preprint arXiv:2406.01499, 2024

arXiv 2024

[7] [7]

Jeff Cooper and Dhruv Mubayi, Coloring sparse hypergraphs, SIAM Journal on Discrete Mathematics 30(2) (2016), 1165–1180

work page 2016

[8] [8]

Henry Dudeney, A puzzle with pawns, in Amusements in Mathematics , Nelson, Edinburgh, 1917, pp. 94

work page 1917