Pith. sign in

REVIEW 2 major objections 2 minor 2 cited by

On a hypergraph Mantel theorem

T0 review · 2 major / 2 minor · reviewed 2026-08-09 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves that for each r ≥ 2, every sufficiently large triangle-free r-graph with minimum degree within o(n^{r-1}) of the Turán threshold is r-partite, making the balanced complete r-partite hypergraph the unique extremal…

desk verdict A plausible and important stability theorem for hypergraph triangles, but the proof as written has two gaps—one in the vertex-extendability step and one in the homomorphic-deletion step—that need repair before the result is established. read the letter →

arxiv 2501.19229 v1 pith:33FY3SHA submitted 2025-01-31 math.CO

Xizhi Liu This is my paper · ORCID
classification math.CO MSC 05C6505C35
keywords hypergraphTuránproblemtriangle-freehypergraphsstabilitytheoremLagrangianentropydegreeextremalAndrásfai–Erdős–Sós
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

The paper proves an Andrásfai–Erdős–Sós-type stability theorem for triangle-free r-uniform hypergraphs: for every r ≥ 2, any sufficiently large hypergraph that avoids the family Δ_r and has minimum degree close to (1/r)^{r-1} $n^{{r-1}}$ must be r-partite. The direct consequence is that, for large n, the balanced complete r-partite r-graph T_r(n) is the unique extremal construction among Δ_r-free hypergraphs on n vertices. This answers an open problem about weakly triangle-free r-graphs in a stronger form than was asked. The proof gets its key input from an entropic reformulation of the hypergraph Lagrangian and from a general stability framework that upgrades edge stability to degree stability.

What carries the argument

The load-bearing object is the hypergraph Lagrangian, the maximum of the polynomial P_H(x) = Σ_{e∈H} ∏_{i∈e} x_i over the simplex, together with its entropic reformulation: the entropy density equals r! λ(H). Proposition 3.1 shows that for a 2-covered triangle-free r-graph the only optimal weight vector is uniform on a single edge, (1/r,...,1/r,0,...,0). This uniqueness is combined with vertex-extendability of the triangle configuration with respect to r-partite hypergraphs and a stability framework that derives degree stability from edge stability plus vertex-extendability.

What would settle it

Exhibit, for some r ≥ 4, a Δ_r-free r-graph on arbitrarily large n with minimum degree at least $n^{{r-1}}$/$r^{{r-1}}$ − o($n^{{r-1}}$) that is not r-partite; this would directly contradict Theorem 1.2. A cheaper check is to compute the Lagrangian of a candidate 2-covered T_r-free hypergraph and see whether it exceeds 1/r^r, which would falsify Proposition 3.1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: there exist ε(r) > 0 and N0(r) such that every Δ_r-free r-graph H on n ≥ N0(r) vertices with δ(H) ≥ $n^{{r-1}}$/$r^{{r-1}}$ − ε $n^{{r-1}}$ is r-partite. For large n this makes T_r(n) the unique extremal Δ_r-free construction, and the same argument yields the exact bound |H| ≤ n^r/r^r, with equality exactly for T_r(n) when r divides n. The theorem is stronger than the weakly triangle-free problem it answers, because Δ_r-freeness is a weaker hypothesis than weakly triangle-free.

Load-bearing premise

The whole argument leans on an imported result that has not been re-proved here: every triangle-free hypergraph has a certain maximum-polynomial value equal to 1/r^r; if that result were wrong, the degree-stability conclusion would not follow.

Editorial extensions

If this is right

  • For large n, T_r(n) is the unique extremal Δ_r-free construction, settling the large-n case of Mubayi–Pikhurko's Problem 20.
  • Combined with a standard blow-up argument, it gives the exact Turán number ex(n, Δ_r) = n^r/r^r for large n, with equality only for T_r(n).
  • Together with known results on spectral Turán problems, it solves the α-spectral Turán problem for Δ_r for all α ≥ 1 and large n.
  • It yields a generalized Turán theorem for copies of Steiner triple systems in C3-free 3-graphs: the extremal number is |T_k(n)|.
  • The same stability route applies to the single hypergraph described in [CY24, Theorem 1.5] with only minor modifications.

Reading between the lines

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

  • Editorial inference: the Lagrangian-uniqueness proposition likely holds for other 'sharp' hypergraph families, so the same proof scheme should yield degree-stability theorems for families whose Lagrangian is known to be 1/r^r and uniquely attained on one edge.
  • Editorial inference: the optimal stability window ε(r) is left open; a natural conjecture is that for r=3 the largest ε matches the exact threshold from the recently solved case, so the stability regime may be as wide as possible.
  • Editorial inference: since the proof only needs the Lagrangian value and uniqueness, the method may transfer to L-intersecting families with L = [i] and i < ⌈r/2⌉, where the same Lagrangian bound holds, potentially answering part of Problem 6.2.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper proves an Andrásfai–Erdős–Sós-type stability theorem for triangle-free r-uniform hypergraphs: for each r ≥ 2, every sufficiently large Δ_r-free r-graph on n vertices with minimum degree at least n^{r-1}/r^{r-1} − ε n^{r-1} is r-partite, provided ε and n are chosen appropriately. It follows that for large n the balanced complete r-partite r-graph T_r(n) is the unique extremal Δ_r-free construction. This is presented as a stronger form of Mubayi and Pikhurko's Problem 20 on weakly triangle-free r-graphs. The proof combines the entropy–Lagrangian framework of Chao–Yu with the stability framework of Liu–Mubayi–Reiher and Hou–Liu–Zhao, together with a new vertex-extendability argument for T_{r,1} and a symmetrized stability proposition.

Significance. If the proof is completed, the result is significant: it gives the first degree-stability statement for the family Δ_r and settles the extremal uniqueness question for weakly triangle-free r-graphs for large n, in a stronger form than originally asked. The paper is clearly organized, and the entropy computations in Section 3 are carefully executed; Proposition 3.1, in particular, is a clean and nontrivial structural statement. The paper is honest about its reliance on external results, especially Chao–Yu's Lagrangian theorem for T_r-free hypergraphs and the stability framework of [LMR23, HLZ24]. However, two load-bearing steps in the proof of Theorem 1.2 are not justified as written.

major comments (2)
  1. [Section 5, final paragraph] The claim that a standard application of the Hypergraph Removal Lemma turns Δ_r-freeness into T_r-freeness after deleting o(n^r) edges is not justified. The usual removal lemma controls subgraph copies of a fixed family, while Δ_r-freeness only forbids subgraphs isomorphic to members of Δ_r, not homomorphisms from those members. A copy of F ∈ T_r in a Δ_r-free graph need not contain any member of Δ_r; for example, for r = 3 the 3-graph F = {abc, abd, acd} belongs to T_3 but has no Δ_3 subgraph, and a star centered at one vertex is Δ_3-free while containing many such F-copies. The text does not supply the required bound on the number of copies of each F in a Δ_r-free graph, nor does it state and verify a homomorphism removal lemma. This step is load-bearing because it converts Δ_r edge-stability into T_r edge-stability before Theorem 2.3(ii) is applied.
  2. [Section 5, proof of Theorem 1.2] Proposition 5.1 is applied to G = H − v* solely because G belongs to the family H. But H is defined as the family of r-graphs that admit a homomorphism to some 2-covered T_r-free r-graph; this does not imply that G is symmetrized, i.e., a full blowup of a 2-covered graph. Proposition 5.1 has symmetrized as an explicit hypothesis. Without an additional argument (for example, a symmetrization procedure preserving the high minimum degree and T_r-freeness), the conclusion that G is r-partite does not follow. This gap affects the proof that T_r is vertex-extendable with respect to H, which is needed for the invocation of Theorem 2.3(i).
minor comments (2)
  1. [Throughout] There are a few harmless typographical slips, such as 'max' used instead of a set in the definition of S_{n−1} in Section 2, and 'Proposition 1.2' instead of 'Theorem 1.2' at the end of the proof in Section 5.
  2. [Section 2, Fact 2.2(ii)] The fact that a 2-covered T_r-free r-graph is a (v(H), r, r−1)-system is used repeatedly; it may be worth adding a one-sentence proof or a precise reference, since it is a key structural input.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the target theorem is not an input; self-cited framework theorems are general criteria, not target-specific assumptions. A removal-lemma bridging step is a correctness risk, not circularity.

full rationale

The paper's central derivation is not circular. Proposition 3.1 proves uniqueness of the Lagrangian optimizer for 2-covered T_r-free graphs using the external Theorem 2.6 from Chao-Yu and the entropy identities proved as Proposition 2.7; Proposition 5.1 converts this into symmetrized degree stability; Theorem 2.3 is a general stability criterion from [LMR23, HLZ24], not a theorem about triangles, so citing it is independent support rather than a self-citation chain that forces the conclusion. The final step of the proof of Theorem 1.2 asserts that a standard application of the Hypergraph Removal Lemma turns every Delta_r-free graph into a T_r-free graph by deleting o(n^r) edges, because every F in T_r is a homomorphic image of some tilde-F in Delta_r. This is a load-bearing inference, but it is not circular: the cited removal lemmas control subgraph copies, and the paper gives no argument bounding homomorphic images. That is a correctness gap, not a reduction of the theorem to its own input. The self-citations in the framework theorems are transparent and do not smuggle in the target result, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on three external mathematical inputs: the entropy-Lagrangian theorem of Chao-Yu, the stability framework of LMR23/HLZ24, and the hypergraph removal lemma. None of these are newly proved here. No free parameters are fitted to data; the constants ε, ε1, ε2, δ are existential and chosen sufficiently small. No new entities are introduced.

assumptions (3)
  • domain assumption Chao-Yu Theorem 2.6: every T_r-free r-graph H has Lagrangian λ(H) = 1/r^r.
    Cited from [CY24], used in Proposition 3.1 to fix β = r!/r^r and in Claim 3.5 to bound the Lagrangian of links. This is the main external input the paper does not re-derive.
  • domain assumption LMR23/HLZ24 Theorem 2.3: a blowup-invariant family that is symmetrized-stable and vertex-extendable with respect to a hereditary family is degree-stable, and edge-stability plus vertex-extendability implies degree-stability.
    Cited from [LMR23, Theorem 1.7] and [HLZ24, Theorem 1.1], this framework is the backbone of the proof of Theorem 1.2. The paper does not prove it.
  • standard math Hypergraph Removal Lemma.
    Used in the final paragraph of the proof of Theorem 1.2 to show that every Δ_r-free r-graph can be made T_r-free by deleting o(n^r) edges, yielding edge-stability for Δ_r.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On a hypergraph Mantel theorem." pith.science (2026). https://pith.science/paper/33FY3SHA

@misc{pith2026250119229,
  author       = {Pith},
  title        = {Pith review of: On a hypergraph Mantel theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/33FY3SHA}},
  note         = {Machine review of arXiv:2501.19229}
}
abstract

An $r$-graph is a triangle if there exists a positive integer $i \le \lceil r/2 \rceil$ such that it is isomorphic to the following $r$-graph with three edges: \begin{align*} \left\{\{1, \ldots, r\},~\{1, \ldots, i, r+1, \ldots, 2r-i\},~\{i+1, \ldots, r, r+1, 2r-i+1, \ldots,2r-1\}\right\}. \end{align*} We prove an Andr{\'a}sfai--Erd\H{o}s--S\'{o}s-type stability theorem for triangle-free $r$-graphs. In particular, it implies that for large $n$, the unique extremal triangle-free construction on $n$ vertices is the balanced complete $r$-partite $r$-graph. The latter result answers a question by Mubayi and Pikhurko~{\cite[Problem~20]{MPS11}} on weakly triangle-free $r$-graphs for large $n$ in a stronger form. The proof combines the recently introduced entropic technique of Chao--Yu~\cite{CY24} with the framework developed in~\cite{LMR23unif,HLZ24}.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Spectral generalized Tur\'{a}n problems

    math.CO 2025-07 conditional novelty 6.0 of 10

    The paper introduces spectral generalized Turán numbers, proves a general transfer theorem from counting stability to spectral extremality, and derives a spectral Erdős Pentagon Theorem and an entropy formula.

  2. Entropy methods in combinatorics

    math.CO 2026-07 accept novelty 2.0 of 10

    A selective survey of entropy methods in combinatorics, detailing randomized chain rules, Shearer's inequality, random homomorphisms, Pinsker-type arguments, the union-closed sets breakthrough, and entropy approaches ...

Reference graph

Works this paper leans on

40 extracted references · 22 canonical work pages · cited by 2 Pith papers

  1. [1]

    Andr\' a sfai, P

    B. Andr\' a sfai, P. Erd o s, and V. T. S\' o s. On the connection between chromatic number, maximal clique and minimal degree of a graph. Discrete Math. , 8:205--218, 1974

    1974

  2. [2]

    Many T copies in H -free graphs

    Noga Alon and Clara Shikhelman. Many T copies in H -free graphs. J. Combin. Theory Ser. B , 121:146--172, 2016

    2016

  3. [3]

    On the chromatic thresholds of hypergraphs

    J \'o zsef Balogh, Jane Butterfield, Ping Hu, John Lenz, and Dhruv Mubayi. On the chromatic thresholds of hypergraphs. Combin. Probab. Comput. , 25(2):172--212, 2016

    work page 2016

  4. [4]

    Three-graphs without two triples whose symmetric difference is contained in a third

    B\' e la Bollob\' a s. Three-graphs without two triples whose symmetric difference is contained in a third. Discrete Math. , 8:21--24, 1974

    work page 1974

  5. [5]

    Strong stability from vertex-extendability and applications in generalized T ur \' a n problems

    Wanfang Chen and Xizhi Liu. Strong stability from vertex-extendability and applications in generalized T ur \' a n problems. arXiv preprint arXiv:2406.05748 , 2024

    arXiv 2024

  6. [6]

    When entropy meets T ur\' a n: new proofs and hypergraph T ur\' a n results

    Ting-Wei Chao and Hung-Hsun Hans Yu. When entropy meets T ur\' a n: new proofs and hypergraph T ur\' a n results. arXiv preprint arXiv:2412.08075 , 2024

    arXiv 2024

  7. [7]

    Jean Doyen and Richard M. Wilson. Embeddings of S teiner triple systems. Discrete Math. , 5:229--239, 1973

    work page 1973

  8. [8]

    P. Erd o s. \" U ber ein E xtremalproblem in der G raphentheorie. Arch. Math. (Basel) , 13:222--227, 1962

    1962

Show all 40 references
  1. [9]

    Erd o s and A

    P. Erd o s and A. H. Stone. On the structure of linear graphs. Bull. Amer. Math. Soc. , 52:1087--1091, 1946

    1946

  2. [10]

    Erd o s and M

    P. Erd o s and M. Simonovits. A limit theorem in graph theory. Studia Sci. Math. Hungar. , 1:51--57, 1966

    1966

  3. [11]

    A new generalization of the E rd o s- K o- R ado theorem

    Peter Frankl and Zolt\'an F \"u redi. A new generalization of the E rd o s- K o- R ado theorem. Combinatorica , 3(3-4):341--349, 1983

    1983

  4. [12]

    Frankl and Z

    P. Frankl and Z. F\" u redi. Extremal problems whose solutions are the blowups of the small W itt-designs. J. Combin. Theory Ser. A , 52(1):129--147, 1989

    1989

  5. [13]

    Frankl and V

    P. Frankl and V. R\" o dl. Hypergraphs do not jump. Combinatorica , 4(2-3):149--159, 1984

    1984

  6. [14]

    Invitation to intersection problems for finite sets

    Peter Frankl and Norihide Tokushige. Invitation to intersection problems for finite sets. J. Combin. Theory Ser. A , 144:157--211, 2016

    2016

  7. [15]

    On the T ur \'a n number of \ 123,124,345\

    John Goldwasser. On the T ur \'a n number of \ 123,124,345\ . Manuscript

  8. [16]

    W. T. Gowers. Hypergraph regularity and the multidimensional S zemer\'edi theorem. Ann. of Math. (2) , 166(3):897--946, 2007

    2007

  9. [17]

    A criterion for A ndr \' a sfai-- E rd o s-- S \' o s type theorems and applications

    Jianfeng Hou, Xizhi Liu, and Hongbin Zhao. A criterion for A ndr \' a sfai-- E rd o s-- S \' o s type theorems and applications. arXiv preprint arXiv:2401.17219 , 2024

    2024 arXiv

  10. [18]

    Hypergraph T ur\' a n problems

    Peter Keevash. Hypergraph T ur\' a n problems. In Surveys in combinatorics 2011 , volume 392 of London Math. Soc. Lecture Note Ser. , pages 83--139. Cambridge Univ. Press, Cambridge, 2011

    2011

  11. [19]

    Spectral extremal problems for hypergraphs

    Peter Keevash, John Lenz, and Dhruv Mubayi. Spectral extremal problems for hypergraphs. SIAM J. Discrete Math. , 28(4):1838--1854, 2014

    2014

  12. [20]

    Stability theorems for cancellative hypergraphs

    Peter Keevash and Dhruv Mubayi. Stability theorems for cancellative hypergraphs. J. Combin. Theory Ser. B , 92(1):163--175, 2004

    2004

  13. [21]

    On a problem of T ur\' a n in the theory of graphs

    Gyula Katona, Tibor Nemetz, and Mikl\' o s Simonovits. On a problem of T ur\' a n in the theory of graphs. Mat. Lapok , 15:228--238, 1964

    1964

  14. [22]

    L. Kang, V. Nikiforov, and X. Yuan. The p -spectral radius of k -partite and k -chromatic uniform hypergraphs. Linear Algebra Appl. , 478:81--107, 2015

    2015

  15. [23]

    New short proofs to some stability theorems

    Xizhi Liu. New short proofs to some stability theorems. European J. Combin. , 96:Paper No. 103350, 8, 2021

    2021

  16. [24]

    Cancellative hypergraphs and S teiner triple systems

    Xizhi Liu. Cancellative hypergraphs and S teiner triple systems. J. Combin. Theory Ser. B , 167:303--337, 2024

    2024

  17. [25]

    The feasible region of hypergraphs

    Xizhi Liu and Dhruv Mubayi. The feasible region of hypergraphs. J. Comb. Theory, Ser. B , 148:23--59, 2021

    2021

  18. [26]

    A unified approach to hypergraph stability

    Xizhi Liu, Dhruv Mubayi, and Christian Reiher. A unified approach to hypergraph stability. J. Combin. Theory Ser. B , 158:36--62, 2023

    2023

  19. [27]

    A ndr \' a sfai-- E rd o s-- S \' o s theorem for the generalized triangle

    Xizhi Liu, Sijie Ren, and Jian Wang. A ndr \' a sfai-- E rd o s-- S \' o s theorem for the generalized triangle. arXiv preprint arXiv:2410.20832 , 2024

    2024 arXiv

  20. [28]

    Positive codegree A ndr \' a sfai-- E rd o s-- S \' o s theorem for the generalized triangle

    Xizhi Liu, Sijie Ren, and Jian Wang. Positive codegree A ndr \' a sfai-- E rd o s-- S \' o s theorem for the generalized triangle. arXiv preprint arXiv:2411.07090 , 2024

    2024 arXiv

  21. [29]

    Vraagstuk XXVIII

    Willem Mantel. Vraagstuk XXVIII . Wiskundige Opgaven , 10(2):60--61, 1907

    1907

  22. [30]

    Hypergraph T ur \'a n problem: some open questions

    Dhruv Mubayi, Oleg Pikhurko, and Benny Sudakov. Hypergraph T ur \'a n problem: some open questions. In AIM workshop problem lists, manuscript , page 166, 2011

    2011

  23. [31]

    T. S. Motzkin and E. G. Straus. Maxima for graphs and a new proof of a theorem of T ur\' a n. Canadian J. Math. , 17:533--540, 1965

    1965

  24. [32]

    The counting lemma for regular k -uniform hypergraphs

    Brendan Nagle, Vojt e ch R \"o dl, and Mathias Schacht. The counting lemma for regular k -uniform hypergraphs. Random Structures Algorithms , 28(2):113--179, 2006

    2006

  25. [33]

    Norin and L

    S. Norin and L. Yepremyan. Tur \'a n number of generalized triangles. J. Combin. Theory Ser. A , 146:312--343, 2017

    2017

  26. [34]

    An exact T ur\' a n result for the generalized triangle

    Oleg Pikhurko. An exact T ur\' a n result for the generalized triangle. Combinatorica , 28(2):187--208, 2008

    2008

  27. [35]

    Regularity lemma for k -uniform hypergraphs

    Vojt e ch R \"o dl and Jozef Skokan. Regularity lemma for k -uniform hypergraphs. Random Structures Algorithms , 25(1):1--42, 2004

    2004

  28. [36]

    C. E. Shannon. A mathematical theory of communication. Bell System Tech. J. , 27:379--423, 623--656, 1948

    1948

  29. [37]

    James B. Shearer. A new construction for cancellative families of sets. Electron. J. Combin. , 3(1):Research Paper 15, approx. 3, 1996

    1996

  30. [38]

    Sidorenko

    A. Sidorenko. An analytic approach to extremal problems for graphs and hypergraphs. In Extremal problems for finite sets ( V isegr\' a d, 1991) , volume 3 of Bolyai Soc. Math. Stud. , pages 423--455. J\' a nos Bolyai Math. Soc., Budapest, 1994

    1991

  31. [39]

    On an extermal problem in graph theory

    Paul Tur \'a n. On an extermal problem in graph theory. Mat. Fiz. Lapok , 48:436--452, 1941

    1941

  32. [40]

    The early history of block designs

    Robin Wilson. The early history of block designs. Rend. Sem. Mat. Messina Ser. II , 9(25):267--276, 2003

    2003

Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.