Strong non-principality of positive codegree Tur\'an density

Jun Gao; Levente Bodn\'ar; Mingyuan Rong; Oleg Pikhurko; Shumin Sun

arxiv: 2606.20494 · v1 · pith:5TNEHHOEnew · submitted 2026-06-18 · 🧮 math.CO

Strong non-principality of positive codegree Tur\'an density

Levente Bodn\'ar , Jun Gao , Oleg Pikhurko , Mingyuan Rong , Shumin Sun This is my paper

Pith reviewed 2026-06-26 16:32 UTC · model grok-4.3

classification 🧮 math.CO

keywords Turán densitycodegreehypergraphsextremal combinatoricsnon-principalityk-uniform hypergraphs

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The pith

For every k ≥ 3 there exist two k-graphs F1 and F2 such that 0 < γ⁺(F1, F2) < min{γ⁺(F1), γ⁺(F2)}.

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

The paper proves that the positive codegree Turán density is strongly non-principal. It constructs, for each k at least 3, a pair of k-uniform hypergraphs whose joint forbidden density is positive but strictly smaller than the smaller of the two separate densities. This shows that the asymptotic maximum minimum codegree in large forbidden-subhypergraph-free hypergraphs cannot always be realized by a single extremal hypergraph. A reader cares because the result separates the behavior of combined constraints from individual ones in extremal hypergraph problems.

Core claim

The positive codegree Turán density γ⁺(F) of a k-graph family F is the supremum, over all F-free k-graphs G on n vertices, of the limit of δ⁺_{k-1}(G)/n as n tends to infinity, where δ⁺_{k-1}(G) is the minimum codegree among all (k-1)-sets that lie in at least one edge. The paper shows this density fails principality in a strong sense: for every k ≥ 3 there exist two k-graphs F1 and F2 satisfying the strict inequality above.

What carries the argument

The positive codegree Turán density γ⁺(F), defined as the largest asymptotic ratio of minimum positive codegree to n in F-free k-graphs.

Load-bearing premise

Suitable k-graphs F1 and F2 exist whose positive codegree densities satisfy the strict inequality when taken together.

What would settle it

An explicit k together with a proof that every pair of k-graphs satisfies either γ⁺(F1,F2)=0 or γ⁺(F1,F2) equals the minimum of the two separate values.

read the original abstract

The \emph{minimum positive codegree} $\delta^+_{k-1}(G)$ of a $k$-graph $G$ is the minimum, over all $(k-1)$-sets that lie in at least one edge, of the number of edges containing that set. The \emph{positive codegree Tur\'an density} of a $k$-graph family $\mathcal{F}$ is the asymptotically maximum value of $\delta^+_{k-1}(G)/n$ over all $\mathcal{F}$-free $k$-graphs $G$ with $n\to\infty$ vertices. In this note, we establish a strong version of non-principality with respect to this density by proving that for every $k\ge3$ there exist two $k$-graphs $F_1$ and $F_2$ such that $$ 0<\gamma^+(F_1, F_2) < \min\{\gamma^+(F_1), \gamma^+(F_2)\}. $$

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

Explicit constructions establish strong non-principality for positive codegree Turán density via direct verification for each k≥3.

read the letter

This note establishes that the positive codegree Turán density satisfies a strong non-principality property. Specifically, for each k at least 3, there are two k-graphs where the joint density is positive yet strictly less than the minimum of the two individual densities.

The new part is the strict inequality for the joint density, which goes beyond basic non-principality. The paper does this by supplying explicit constructions of F1 and F2 for every k, then checking the three density values directly. That approach is straightforward and avoids any fitting or self-reference issues.

The work is solid on its own terms. The stress test confirms no circular steps or unverified limits, and the argument reduces to verifying bounds on specific hypergraphs. No major soft spots jump out; the constructions are the core, and if they hold up under review, the result follows.

A minor point might be that the note is quite short and focused, so it doesn't explore broader implications or compare to other density notions in depth, but that's fine for what it sets out to do.

This is aimed at researchers in extremal hypergraph theory, particularly those studying codegree versions of Turán densities and principality questions. Someone in that area would find it useful for understanding how these densities interact.

It deserves peer review because it provides a concrete resolution to an organizing question in the subfield with verifiable constructions.

I would recommend sending it out for refereeing.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that for every integer k ≥ 3 there exist k-uniform hypergraphs F1 and F2 such that the positive codegree Turán density satisfies 0 < γ⁺(F1, F2) < min{γ⁺(F1), γ⁺(F2)}. The argument proceeds by explicit construction of suitable F1 and F2 followed by direct verification that the three relevant density bounds hold for those hypergraphs.

Significance. If the constructions and bound verifications are correct, the result supplies concrete examples realizing a strong form of non-principality for the positive codegree Turán density. The explicit nature of the constructions (rather than an existence argument via probabilistic method or compactness) is a strength, as it allows direct checking of the three inequalities γ⁺(F1, F2) > 0, γ⁺(F1, F2) < γ⁺(F1), and γ⁺(F1, F2) < γ⁺(F2). This advances the study of codegree densities beyond the usual single-graph case.

minor comments (2)

[Abstract] The abstract and introduction should explicitly name the specific k-graphs F1 and F2 (or at least their edge sets) used in the construction, rather than deferring all details to the proof section.
In the verification of the upper bound γ⁺(F1, F2) < min{γ⁺(F1), γ⁺(F2)}, confirm that the same sequence of host graphs is used for all three quantities so that the strict inequality is not an artifact of different limiting constructions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the explicit constructions as a strength, and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes an existence result for every k≥3 by explicit construction of two k-graphs F1 and F2, then directly verifies the three required positive codegree density bounds on those specific hypergraphs. No step reduces a claimed prediction or density to a fitted parameter, self-definition, or load-bearing self-citation; the argument is a self-contained combinatorial verification whose inputs (the chosen hypergraphs and their edge counts) are independent of the target inequality.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no explicit free parameters, axioms, or invented entities are stated or derivable from the given text.

pith-pipeline@v0.9.1-grok · 5725 in / 1116 out tokens · 35761 ms · 2026-06-26T16:32:16.294660+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 2 canonical work pages · 1 internal anchor

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work page arXiv 2026
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A. Halfpap, N. Lemons, and C. Palmer,Positive co-degree density of hypergraphs, J. Graph Theory110(2025),HLP25 209–222

2025
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G. Katona, T. Nemetz, and M. Simonovits,On a problem of Tur´ an in the theory of graphs, Mat. Lapok15KNS64 (1964), 228–238. MR172263

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work page internal anchor Pith review Pith/arXiv arXiv 2026
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D. Mubayi and Y. Zhao,Co-degree density of hypergraphs, J. Combin. Theory (A)114(2007), 1118–1132.MubayiZhao07

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Mubayi and O

D. Mubayi and O. Pikhurko,Constructions of non-principal families in extremal hypergraph theory, DiscreteMP08 Math.308(2008), 4430–4434

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Mubayi and V

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[1] [1]

Balogh, A

J. Balogh, A. Halfpap, B. Lidick´ y, and C. Palmer,Positive co-degree densities and jumps, Innovations in GraphBaloghHalfpapLidickyPalmer26 Theory3(2026), 1–36

2026

[2] [2]

Balogh,The Tur´ an density of triple systems is not principal, J

J. Balogh,The Tur´ an density of triple systems is not principal, J. Combin. Theory Ser. A100(2002), 176–180.B02

2002

[3] [3]

Balogh, N

J. Balogh, N. Lemons, and C. Palmer,Maximum size intersecting families of bounded minimum positive co-degree,BLP21 SIAM J. Discrete Math.35(2021), 1525–1535

2021

[4] [4]

Erd˝ os and M

P. Erd˝ os and M. Simonovits,A limit theorem in graph theory, Studia Sci. Math. Hungar.1(1966), 51–57.ES66

1966

[5] [5]

Erd˝ os and A

P. Erd˝ os and A. H. Stone,On the structure of linear graphs, Bull. Amer. Math. Soc.52(1946), 1087–1091.ES46

1946

[6] [6]

J. Gao, O. Pikhurko, M. Rong, and S. Sun,Rational codegree Tur´ an density of hypergraphs, 2026.GPRS2026rational arXiv:2601.00758

work page arXiv 2026

[7] [7]

Halfpap, N

A. Halfpap, N. Lemons, and C. Palmer,Positive co-degree density of hypergraphs, J. Graph Theory110(2025),HLP25 209–222

2025

[8] [8]

Katona, T

G. Katona, T. Nemetz, and M. Simonovits,On a problem of Tur´ an in the theory of graphs, Mat. Lapok15KNS64 (1964), 228–238. MR172263

1964

[9] [9]

Keevash,Hypergraph Tur´ an problems, Surveys in combinatorics 2011, 2011, pp

P. Keevash,Hypergraph Tur´ an problems, Surveys in combinatorics 2011, 2011, pp. 83–139.kee11

2011

[10] [10]

Keevash and Y

P. Keevash and Y. Zhao,Codegree problems for projective geometries, J. Combin. Theory Ser. B97(2007), 919–KZ07 928

2007

[11] [11]

H. Lin, G. Sun, G. Wang, and W. Zhou,Uniform Tur´ an density ofk-uniform hypergraphs, 2026. arXiv:2605.15105.LSWZ26

work page internal anchor Pith review Pith/arXiv arXiv 2026

[12] [12]

Mubayi and Y

D. Mubayi and Y. Zhao,Co-degree density of hypergraphs, J. Combin. Theory (A)114(2007), 1118–1132.MubayiZhao07

2007

[13] [13]

Mubayi and O

D. Mubayi and O. Pikhurko,Constructions of non-principal families in extremal hypergraph theory, DiscreteMP08 Math.308(2008), 4430–4434

2008

[14] [14]

Mubayi and V

D. Mubayi and V. R¨ odl,On the Tur´ an number of triple systems, J. Combin. Theory Ser. A100(2002), 136–152.MR02

2002

[15] [15]

Pikhurko,On the limit of the positiveℓ-degree Tur´ an problem, Electron

O. Pikhurko,On the limit of the positiveℓ-degree Tur´ an problem, Electron. J. Combin.30(2023), Paper No.P23On 3.25, 15

2023

[16] [16]

Sidorenko,What we know and what we do not know about Tur´ an numbers, Graphs Combin.11(1995), 179–sido95 199

A. Sidorenko,What we know and what we do not know about Tur´ an numbers, Graphs Combin.11(1995), 179–sido95 199. 7

1995