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arxiv: 2606.04400 · v2 · pith:RW7G5QDWnew · submitted 2026-06-03 · 🧮 math.CO

On linear k-graphs with codegree Tur\'an density arbitrarily close to zero

Xiaona Fang , Yaojun Chen This is my paper

Pith reviewed 2026-06-28 05:52 UTC · model grok-4.3

classification 🧮 math.CO
keywords linear hypergraphscodegree Turán densityhypergraph Turán problemsextremal hypergraph theoryaffine planesfinite fieldsincidence structures
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The pith

There exist linear k-graphs with codegree Turán density positive but arbitrarily close to zero.

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

The paper establishes that for every ε > 0 there is a linear k-uniform hypergraph F such that its codegree Turán density satisfies 0 < π_co(F) < ε. This quantity is the largest γ such that arbitrarily large F-free k-graphs can still have every (k-1)-set in at least γn edges. The result gives a positive answer for k = 3 to a question posed in 2025. The proof proceeds by an explicit construction that keeps the hypergraph linear while driving the density down.

Core claim

The authors prove that there is a linear k-graph F with 0 < π_co(F) < ε for any ε > 0. The construction uses an affine-plane-type incidence structure over a finite field together with elementary number-theoretic arguments to ensure linearity while making the codegree Turán density arbitrarily small.

What carries the argument

An affine-plane-type incidence structure over a finite field, which produces a linear k-graph whose codegree Turán density lies strictly between zero and any prescribed ε.

If this is right

  • For every ε > 0 and every k ≥ 3 there exists at least one linear F with codegree Turán density in (0, ε).
  • The k = 3 case directly resolves the question of Ding, Lamaison, Liu, Wang and Yang.
  • Codegree Turán densities of linear hypergraphs are not bounded away from zero.
  • The same geometric-arithmetic method yields a family of examples with successively smaller positive densities.

Where Pith is reading between the lines

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

  • The possible values of π_co(F) for linear F may be dense in (0,1).
  • One could ask whether every number in (0,1) arises as π_co(F) for some linear F.
  • The construction might be adapted to control other variants of hypergraph Turán densities.
  • It remains open whether the infimum of positive codegree densities over all linear F is zero or some positive constant.

Load-bearing premise

The affine-plane incidence structure over finite fields, combined with number-theoretic arguments, produces a linear hypergraph whose codegree Turán density is bounded above by any given ε.

What would settle it

An explicit computation or counter-example showing that the constructed F is either not linear or has codegree Turán density bounded away from zero by some fixed positive constant independent of ε.

Figures

Figures reproduced from arXiv: 2606.04400 by Xiaona Fang, Yaojun Chen.

Figure 1
Figure 1. Figure 1: O (3) 6,2 The main result of this paper is as follows. Theorem 1.1. For any ε > 0, there is an ℓ ∈ N such that 0 < πco(O (k) ℓ,p ) < ε. Clearly, Theorem 1.1 answers Question 1.1 in affirmative. 2 Preliminaries In this section, we introduce some additional definitions and some known results for the purpose to prove Theorem 1.1. Definition 2.1. Let m ≥ 2 be an integer. We define the n-vertex k-graph H (k) m … view at source ↗
read the original abstract

Let $F$ be a $k$-uniform hypergraph, abbreviated as $k$-graph. The codegree Tur\'an density $\pi_{\mathrm{co}}(F)$ is the supremum over all $\gamma \in [0,1)$ such that, for arbitrarily large $n$, there exists an $n$-vertex $F$-free $k$-graph $H$ whose every $(k-1)$-subset of vertices lies in at least $\gamma n$ edges. In this paper, we prove that there is a linear $k$-graph $F$ with $0<\pi_{co}(F) < \varepsilon$ for any $\varepsilon>0$. The special case $k=3$ solve a question proposed by Ding, Lamaison, Liu, Wang and Yang (JLMS, 2025). The main method combines an affine-plane-type incidence structure over a finite field and elementary number-theoretic arguments.

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

0 major / 3 minor

Summary. The paper proves that for every k ≥ 3 and every ε > 0 there exists a linear k-uniform hypergraph F such that its codegree Turán density satisfies 0 < π_co(F) < ε. The construction relies on an affine-plane-type incidence structure over a finite field together with elementary number-theoretic arguments. The k = 3 case resolves a question posed by Ding, Lamaison, Liu, Wang and Yang (JLMS 2025).

Significance. If the stated construction is correct, the result shows that codegree Turán densities of linear k-graphs can be made arbitrarily small yet positive. This supplies the first examples separating the codegree density from both zero and the ordinary Turán density in a controlled way and answers an explicit open question for triple systems.

minor comments (3)
  1. The abstract and introduction should include a brief comparison of the new codegree densities with the known ordinary Turán densities of the same linear hypergraphs, to clarify how much the codegree version improves upon the classical bound.
  2. Notation for the finite-field incidence structure (e.g., the precise definition of the vertex set and edge set in terms of F_q) should be introduced once in a dedicated subsection rather than scattered across the construction argument.
  3. The number-theoretic estimates used to bound the codegree from below would benefit from an explicit statement of the prime-power size q that works for a given ε; a short table or corollary listing admissible q would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the result (including resolution of the k=3 case), and recommendation for minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; explicit constructive existence proof

full rationale

The paper establishes an existence result for linear k-graphs F with 0 < π_co(F) < ε via an explicit construction that combines affine-plane-type incidence structures over finite fields with elementary number-theoretic arguments. No load-bearing step reduces by definition, fitting, or self-citation to the target claim itself; the derivation relies on independent external combinatorial and algebraic objects rather than internal reparameterization or prior self-referential results. The cited question from Ding et al. (JLMS 2025) is external and does not form a self-citation chain. This is a standard non-circular constructive proof.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence and properties of an affine-plane incidence structure over a finite field that produces the desired linear hypergraph; these are standard combinatorial objects.

axioms (1)
  • standard math Existence and incidence properties of affine planes over finite fields
    Invoked to build the incidence structure that controls codegrees while preserving linearity.

pith-pipeline@v0.9.1-grok · 5691 in / 1117 out tokens · 30580 ms · 2026-06-28T05:52:39.491805+00:00 · methodology

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Reference graph

Works this paper leans on

7 extracted references · 2 canonical work pages

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    work page arXiv 2025

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    S. Piga, B. Schülke, Hypergraphs with arbitrarily small codegree Turán density, Bull. Lond. Math. Soc. 58(4) (2026) e70348. 6

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