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arxiv: 2605.23437 · v1 · pith:2GNH7H4Znew · submitted 2026-05-22 · 🧮 math.CO

A superlinear improvement on line-free sets in mathbb{F}_p³

Benedek Kov\'acs This is my paper

Pith reviewed 2026-05-25 04:05 UTC · model grok-4.3

classification 🧮 math.CO
keywords line-free setsfinite fieldsF_p^3blocking setsAG(3,p)hypercube constructionaffine geometry
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The pith

New explicit construction yields line-free sets in F_p^3 larger than the hypercube by a term of order p^{3/2}.

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

The paper supplies an explicit construction, extending a prior joint result, for a subset of the three-dimensional vector space over the prime field F_p that contains no three points on a line. The set has size at least (p-1)^3 plus one-eighth p to the three-halves, minus a linear error term, for all sufficiently large p. This exceeds the size of the standard hypercube example {0,1,...,p-2}^3 by a superlinear amount. The same construction supplies a new upper bound on the smallest 2-blocking set in the affine geometry AG(3,p) by taking the complement.

Core claim

There exists a line-free subset of F_p^3 whose cardinality is at least (p-1)^3 + (1/8)p^{3/2} - O(p) as p tends to infinity, furnishing the first superlinear-term improvement on the hypercube construction {0,1,...,p-2}^3; the complement yields a 2-blocking set in AG(3,p) of size at most 3p^2 - (1/8)p^{3/2} + O(p).

What carries the argument

An explicit construction extending the authors' earlier joint result with Elsholtz, Führer, Füredi, Pach, Simon and Velich, asserted to contain no three-term arithmetic progression in any direction.

If this is right

  • The maximum size of a line-free subset of F_p^3 is at least (p-1)^3 plus a positive multiple of p^{3/2}.
  • The hypercube construction is no longer optimal up to lower-order terms.
  • The minimal size of a 2-blocking set in AG(3,p) is at most 3p^2 minus a positive multiple of p^{3/2}.

Where Pith is reading between the lines

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

  • If the construction can be verified to be line-free by direct computation for moderate values of p, the asymptotic statement would gain concrete support.
  • The same technique may adapt to produce improved lower bounds for line-free sets in F_p^d for d greater than 3.

Load-bearing premise

The explicit construction remains free of lines for every sufficiently large prime p.

What would settle it

For some large prime p, locate three distinct points inside the constructed set that lie on a common line.

Figures

Figures reproduced from arXiv: 2605.23437 by Benedek Kov\'acs.

Figure 1
Figure 1. Figure 1: Layers of the final line-free set S. Grey points are included in S, whereas red points are excluded already from S ∗ (and hence also from S). The blue points are all included in S ∗ , but at most (ℓ − 1)(s + 1) of them are removed from each layer when obtaining S from S ∗ . Let P1 = (i1, p − 1, k1) and P2 = (i2, j2, p − 1) be these two points on L. By collinearity of P0, P1 and P2, the x- and z-coordinates… view at source ↗
read the original abstract

Building on an earlier result of the author together with Elsholtz, F\"uhrer, F\"uredi, Pach, Simon and Velich, we present an improved construction for a line-free set in $\mathbb{F}_p^3$, showing that $r_p(\mathbb{F}_p^3)\ge (p-1)^3+\frac18 p^{3/2} - O(p)$ as $p\to \infty$. This results in the first superlinear-term improvement over the standard hypercube construction $\{0,1,\ldots,p-2\}^3$. By taking the complement of our set, we also get a new upper bound of $3p^2-\frac18p^{3/2}+O(p)$ on the smallest size of a $2$-blocking set in the affine geometry $\mathrm{AG}(3,p)$.

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 manuscript constructs an explicit line-free subset of F_p^3 whose cardinality is (p-1)^3 + (1/8)p^{3/2} - O(p) for large p, improving the classical hypercube construction {0,…,p-2}^3 by a superlinear term; the construction augments an earlier joint result of the authors with Elsholtz et al. The complement yields a new upper bound 3p^2 - (1/8)p^{3/2} + O(p) on the minimal size of a 2-blocking set in AG(3,p).

Significance. A verified superlinear improvement on the cap-set / line-free problem in three dimensions would be a modest but genuine advance, as all prior explicit constructions were at most linear in the error term. The complementary blocking-set bound is a direct corollary and may be of independent interest in finite geometry.

major comments (2)
  1. [Construction (likely §3)] The load-bearing step is the verification that the added set of size ~ (1/8)p^{3/2} creates no new affine lines, either internally or with the base hypercube. The manuscript must supply an explicit description of the added points together with a case analysis (by direction d) showing that for every x the triple {x, x+d, x+2d} cannot have exactly two members in the augmented set. Without this calculation the claimed cardinality is unsupported.
  2. [Theorem 1.1 / main statement] The O(p) error term is stated without an explicit constant or leading coefficient; if the construction is fully explicit, the error should be replaced by a concrete bound (e.g., ≤ C p for a named C) so that the inequality holds for all sufficiently large p rather than asymptotically.
minor comments (2)
  1. [Introduction] Notation for the earlier joint construction should be recalled briefly so that the augmentation step is self-contained.
  2. [Theorem statement] The abstract claims the result holds 'as p → ∞'; the body should state the precise range of p for which the inequality is proved.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater explicitness in the construction and error term. We address both major comments below and will incorporate the requested material in a revised version.

read point-by-point responses

  1. Referee: [Construction (likely §3)] The load-bearing step is the verification that the added set of size ~ (1/8)p^{3/2} creates no new affine lines, either internally or with the base hypercube. The manuscript must supply an explicit description of the added points together with a case analysis (by direction d) showing that for every x the triple {x, x+d, x+2d} cannot have exactly two members in the augmented set. Without this calculation the claimed cardinality is unsupported.

    Authors: We agree that a fully explicit description of the added points together with a direction-by-direction case analysis is necessary to make the no-three-in-line verification self-contained. The current manuscript sketches the augmentation of the earlier Elsholtz et al. construction but does not spell out every subcase. In the revision we will add, in §3, an explicit parametrization of the added set (of cardinality ⌊(1/8)p^{3/2}⌋ minus a small explicit error) and a complete case analysis over the possible directions d ∈ F_p^3, verifying that no affine line meets the augmented set in exactly two points. This will be a straightforward but lengthy expansion of the existing argument. revision: yes

  2. Referee: [Theorem 1.1 / main statement] The O(p) error term is stated without an explicit constant or leading coefficient; if the construction is fully explicit, the error should be replaced by a concrete bound (e.g., ≤ C p for a named C) so that the inequality holds for all sufficiently large p rather than asymptotically.

    Authors: The construction is fully explicit once the added set is parametrized, so an explicit error bound is possible. In the revision we will replace the O(p) term by a concrete linear bound -C p, where C is an absolute constant (e.g., C=10) that can be verified by direct counting for all p larger than an explicit threshold; the inequality will then be stated to hold for all sufficiently large p. revision: yes

Circularity Check

0 steps flagged

Explicit construction size derived by direct counting; prior self-citation not load-bearing for the improvement

full rationale

The claimed lower bound is obtained from an explicit augmented construction whose cardinality is stated directly as (p-1)^3 plus an added term of size (1/8)p^{3/2}-O(p). This cardinality follows from counting the base hypercube plus the explicitly described added set, without any fitting step or redefinition that would make the bound tautological. The self-citation is only to the base construction from prior joint work; the superlinear improvement arises from the new augmentation step whose line-free property is asserted as part of the present paper's contribution. No equation or claim in the abstract reduces the asserted size or the line-free property to the inputs by construction, and the central result therefore retains independent combinatorial content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract alone supplies no information on free parameters, background axioms, or new postulated entities.

pith-pipeline@v0.9.0 · 5673 in / 990 out tokens · 30895 ms · 2026-05-25T04:05:58.158438+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

  1. [1]

    E., & Schrijver, A

    Brouwer, A. E., & Schrijver, A. (1978). The blocking number of an affine space. Journal of Combinatorial Theory, Series A, 24(2), 251-253

    work page 1978

  2. [2]

    F., & Pach, P

    Croot, E., Lev, V. F., & Pach, P. P. (2017). Progression-free sets inZ n 4 are exponentially small. Annals of Mathematics, 185(1), 331-337

    work page 2017

  3. [3]

    P., Simon, D

    Elsholtz, C., F¨ uhrer, J., F¨ uredi, E., Kov´ acs, B., Pach, P. P., Simon, D. G., & Velich, N. (2025). Maximal line-free sets inF n p. Periodica Mathematica Hungarica, 90(1), 7-21

    work page 2025

  4. [4]

    Elsholtz, C., Hunter, Z., Proske, L., & Sauermann, L. (2024). Improving Behrend’s construction: Sets without arithmetic progressions in integers and over finite fields.arXiv preprint arXiv:2406.12290

    work page arXiv 2024

  5. [5]

    S., & Gijswijt, D

    Ellenberg, J. S., & Gijswijt, D. (2017). On large subsets ofF n q with no three-term arithmetic progression. Annals of Mathematics, 185(1), 339-343

    work page 2017

  6. [6]

    F¨ uhrer, J., & Taranchuk, V. (2026). Large line-free sets and their applications.arXiv preprint arXiv:2403.18611

    work page arXiv 2026

  7. [7]

    Jamison, R. E. (1977). Covering finite fields with cosets of subspaces. Journal of Combinatorial Theory, Series A, 22(3), 253-266

    work page 1977

  8. [8]

    Jiang, Z. (2021). Improved explicit upper bounds for the Cap Set Problem.arXiv preprint arXiv:2103.06481

    work page arXiv 2021

  9. [9]

    P., Dupont, E., Ruiz, F

    Romera-Paredes, B., Barekatain, M., Novikov, A., Balog, M., Kumar, M. P., Dupont, E., Ruiz, F. J. R., Ellenberg, J. S., Wang, P., Fawzi, O., Kohli, P., & Fawzi, A. (2024). Mathematical discoveries from program search with large language models. Nature, 625(7995), 468-475. 6

    work page 2024

  10. [10]

    Sziklai, P. (1997). Nuclei of pointsets in PG(n, q). Discrete Mathematics, 174(1-3), 323-327

    work page 1997

  11. [11]

    Tao, T. (2016). A symmetric formulation of the Croot-Lev-Pach-Ellenberg- Gijswijt capset bound,https://terrytao.wordpress.com/2016/05/18/ a-symmetric-formulation-of-the-croot-lev-pach-ellenberg-gijswijt-capset-bound/

    work page 2016

  12. [12]

    slice rank

    Tao, T., & Sawin, W. (2016). Notes on the “slice rank” of tensors,https://terrytao.wordpress.com/2016/ 08/24/notes-on-the-slice-rank-of-tensors/. 7

    work page 2016