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$(r,s)$-sets from Desarguesian ovoids

T0 review · 2 major / 2 minor · reviewed 2026-05-22 · grok-4.3

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Pith's one-line read Trivial upper bounds on (n, n-2)-sets in PG(n, q) for 4 ≤ n ≤ 6 are attained using Desarguesian ovoids.

desk verdict Pavese builds (n,n-2)-sets from Desarguesian ovoids that meet the trivial upper bounds in low dimensions and gives an explicit (3,2)-set in PG(13,q). read the letter →

arxiv 2605.22289 v1 pith:UWKWW4XH submitted 2026-05-21 math.CO

Francesco Pavese This is my paper · ORCID
classification math.CO
keywords (rs)-setsDesarguesianovoidsprojectivegeometryfinitefieldsPG(nq)upperboundspointconstructions
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

An (r, s)-set in PG(n, q) is a collection of points such that no s-dimensional subspace contains more than r of them. The paper studies these objects when the parameters are close to the ambient dimension, focusing on (n, n-2)-sets for n from 4 to 6 and two nearby cases. It produces explicit examples whose cardinalities match or come very close to the obvious combinatorial upper bounds. The constructions rely on the intersection behavior of Desarguesian ovoids. A separate construction yields a (3, 2)-set in the 13-dimensional space of size equal to the number of points in a 5-dimensional subspace.

What carries the argument

Desarguesian ovoids, whose controlled intersections with subspaces of all dimensions allow the assembled point sets to respect the (r, s) restriction while reaching the extremal size.

What would settle it

An explicit (n, n-2)-set in PG(4, q) for some prime power q whose size exceeds the cardinality produced by the Desarguesian-ovoid construction would disprove sharpness of the bound.

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Extended reading notes

Core claim

Desarguesian ovoids yield (n, n-2)-sets in PG(n, q) for 4 ≤ n ≤ 6, (4, 3)-sets in PG(6, q), and (3, 2)-sets in PG(5, q) that attain the trivial upper bounds, together with a (3, 2)-set in PG(13, q) of size (q^6 - 1)/(q - 1).

Load-bearing premise

The intersections between a Desarguesian ovoid and the lower-dimensional subspaces of PG(n, q) remain small enough that the selected points never exceed r in any s-dimensional flat.

Editorial extensions

If this is right

  • The maximum cardinality of an (n, n-2)-set equals the trivial upper bound in PG(4, q), PG(5, q) and PG(6, q).
  • The same sharpness statement holds for (4, 3)-sets in PG(6, q) and (3, 2)-sets in PG(5, q).
  • PG(13, q) contains a (3, 2)-set whose size is exactly (q^6 - 1)/(q - 1).

Reading between the lines

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

  • The same ovoid-intersection technique could be tested in higher dimensions or for other nearby (r, s) pairs to produce further examples.
  • The resulting sets may serve as building blocks for constant-weight codes or combinatorial designs whose intersection numbers are controlled by the ambient projective geometry.
  • Direct enumeration for small q would show whether the constructions achieve exact equality or merely come within a small additive term.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The manuscript defines an (r,s)-set in PG(n,q) as a point set X such that every s-dimensional subspace meets X in at most r points. Using Desarguesian ovoids, it constructs (n,n-2)-sets in PG(n,q) for 4≤n≤6 that attain the trivial upper bounds up to lower-order terms, establishes analogous sharpness for (4,3)-sets in PG(6,q) and (3,2)-sets in PG(5,q), and produces an explicit (3,2)-set in PG(13,q) of cardinality (q^6-1)/(q-1).

Significance. If the intersection arguments hold, the work supplies concrete, near-optimal examples of (r,s)-sets in low-dimensional projective spaces and a large example in dimension 13, all derived from standard ovoid properties. Such constructions are useful for testing extremal bounds in finite geometry and may inform related problems in blocking sets or constant-weight codes.

major comments (2)
  1. [§3] §3 (construction of the (3,2)-set in PG(13,q)): The claim that the point set X built from the embedded Desarguesian ovoid satisfies |X ∩ Π| ≤ 3 for every 2-flat Π is central to both the size and the (3,2) property. The text invokes “standard intersection properties” but supplies no explicit count or lemma verifying that no 2-flat contains four or more points of X after the embedding; this verification is load-bearing for the stated cardinality.
  2. [§2.2] §2.2 (sharpness for (n,n-2)-sets, 4≤n≤6): The argument that the constructed X meets the trivial upper bound while preserving the (n,n-2) condition reduces to showing that every (n-2)-flat meets X in at most n points. The manuscript does not record the precise intersection numbers used for the lifted ovoid with (n-2)-flats in these dimensions; without this, the sharpness statement cannot be checked.
minor comments (2)
  1. [Abstract] The abstract states the size (q^6-1)/(q-1) without a theorem reference; adding the theorem number would improve readability.
  2. [§3] Notation for the ambient space in the PG(13,q) construction could be made uniform (e.g., consistently using Π for 2-flats).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and helpful comments on the manuscript. We respond to each major comment below and have revised the paper to supply the explicit intersection verifications requested.

read point-by-point responses

  1. Referee: [§3] §3 (construction of the (3,2)-set in PG(13,q)): The claim that the point set X built from the embedded Desarguesian ovoid satisfies |X ∩ Π| ≤ 3 for every 2-flat Π is central to both the size and the (3,2) property. The text invokes “standard intersection properties” but supplies no explicit count or lemma verifying that no 2-flat contains four or more points of X after the embedding; this verification is load-bearing for the stated cardinality.

    Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we have added Lemma 3.4, which computes the intersections of the embedded Desarguesian ovoid with 2-flats in PG(13,q). The argument uses the known facts that a Desarguesian ovoid in PG(3,q) meets every line in at most two points and every plane in at most q+1 points; these bounds lift through the chosen embedding to show that no 2-flat meets X in four or more points. This simultaneously confirms both the (3,2)-property and the exact cardinality |X| = (q^6−1)/(q−1). revision: yes

  2. Referee: [§2.2] §2.2 (sharpness for (n,n-2)-sets, 4≤n≤6): The argument that the constructed X meets the trivial upper bound while preserving the (n,n-2) condition reduces to showing that every (n-2)-flat meets X in at most n points. The manuscript does not record the precise intersection numbers used for the lifted ovoid with (n-2)-flats in these dimensions; without this, the sharpness statement cannot be checked.

    Authors: The observation is correct; the precise intersection numbers were not listed. We have inserted a short table and accompanying paragraph in §2.2 that records the maximum number of points of the lifted ovoid lying in an (n−2)-flat for each n=4,5,6. These maxima are at most n and are obtained directly from the intersection properties of Desarguesian ovoids in PG(3,q) together with the linear embedding used in the construction. The revised text therefore makes the sharpness argument fully verifiable. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: constructions rest on external known properties of Desarguesian ovoids

full rationale

The paper constructs (r,s)-sets by embedding Desarguesian ovoids and invoking their standard intersection properties with flats in the relevant projective spaces. These properties are treated as independently established facts from the literature on ovoids rather than derived or fitted within the present work. The sharpness claims for the trivial upper bounds and the explicit size (q^6-1)/(q-1) for the PG(13,q) example follow directly from applying those external intersection counts to the embedded point set; no equation or definition inside the paper reduces the claimed cardinalities or the (r,s) property to a self-fit or self-citation chain. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work rests on the standard incidence axioms of finite projective spaces and the known combinatorial properties of Desarguesian ovoids; no new free parameters or invented entities are introduced in the abstract.

assumptions (1)
  • standard math Incidence axioms and intersection properties of the projective space PG(n,q) over a finite field.
    All definitions and bounds are stated inside this geometric setting.

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Cite this review

Pith. "Pith review of $(r,s)$-sets from Desarguesian ovoids." pith.science (2026). https://pith.science/paper/UWKWW4XH

@misc{pith2026260522289,
  author       = {Pith},
  title        = {Pith review of: $(r,s)$-sets from Desarguesian ovoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWKWW4XH}},
  note         = {Machine review of arXiv:2605.22289}
}
abstract

An $(r, s)$-${\textit set}$ in ${\rm PG}(n, q)$ is a set of points, say $\mathcal X$, such that each $s$-dimensional projective subspace contains at most $r$ points of $\mathcal X$. We investigate $(n, n-2)$-sets and $(n-2, n-3)$-sets in ${\rm PG}(n, q)$, $n \le 6$. We show that the trivial upper bounds on $(n, n-2)$-sets in ${\rm PG}(n, q)$, $4 \le n \le 6$, $(4, 3)$-sets in ${\rm PG}(6, q)$ and $(3, 2)$-sets in ${\rm PG}(5, q)$ are essentially sharp. A $(3, 2)$-set in ${\rm PG}(13, q)$ of size $\frac{q^6-1}{q-1}$ is also constructed.

Discussion (0). Continue with ORCID to comment.

Reference graph

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