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REVIEW 3 major objections 4 minor 34 references

Tur\'{a}n density of tight cycles minus one edge in the $\ell_2$-norm

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper proves that the $\ell_2$-norm Tur\'an density of $C_\ell^{3-}$ is $1/26$ for every $\ell \ge 5$ with $\ell$ not divisible by $3$, together with an Erd\H{o}s--Simonovits-type stability theorem.

desk verdict The conjectured 1/26 density is settled in a stronger form, but the proof's load-bearing Proposition 3.2 is omitted, so the result is conditional on unverified strengthening. read the letter →

arxiv 2507.00812 v1 pith:33NANWEC submitted 2025-07-01 math.CO

Levente Bodnár , Jinghua Deng , Jianfeng Hou , Xizhi Liu , Hongbin Zhao This is my paper · ORCID
classification math.CO MSC 05C6505D05
keywords Turándensityℓ2-normtightcycle3-uniformhypergraphflagalgebracodegreestabilityTrecconstruction
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 that the $\ell_2$-norm Tur\'an density of the $3$-uniform tight $\ell$-cycle minus one edge is $1/26$ for every $\ell \ge 5$ that is not divisible by $3$. In plain terms, among $n$-vertex hypergraphs that avoid $C_\ell^{3-}$, the sum of squared codegrees is at most $(1/26+o(1))n^4$, and this bound is achieved by the Trec construction, the iterated three-partite blow-up of a single triple. The paper also proves a stability theorem: any near-extremal hypergraph can be turned into a subconstruction of Trec by deleting $o(n^3)$ edges. This confirms, in a stronger form, a conjecture raised in a 2022 survey on $\ell_2$-norm Tur\'an problems. The interest is that a norm sensitive to codegree distribution, not just edge count, still selects the same recursive structure as the extremal family.

What carries the argument

The load-bearing identity is $N(S_2,H)=\sum_{e\in\partial H}\binom{d_H(e)}{2}=(\|H\|_2-3|H|)/2$, which converts the codegree-squared norm into a count of $S_2$ copies. The proof then works on a locally maximal $3$-partition $V_1\cup V_2\cup V_3$ of a $\{K_4^{3-},C_5^{3-}\}$-free hypergraph, comparing bad edges $B$ (edges meeting the parts in pattern $\{0,1,2\}$) with missing transversal triples $M$ via $|B|\le \tfrac34|M|$, and bad $S_2$ copies $B_{S_2}$ with missing $S_2$ copies $M_{S_2}$ via $|B_{S_2}|\le \tfrac9{10}|M_{S_2}|+o(n^4)$. The latter bound comes from a flag algebra semi-definite program over $6$-vertex flags with $16{,}181$ linear constraints. These comparisons feed Lemma 4.1, a recursive inequality $N(S_2,H)\le |V_1||V_2||V_3|n/2+\sum_i N(S_2,H[V_i])+\varepsilon n^4-\max\{|B_{S_2}|/9,|M_{S_2}|/10\}$, whose iteration together with the elementary inequality $x_1x_2x_3/(1-\sum_i x_i^4)\le 1/26$ on the simplex yields the value $6/13$ and the stability theorem.

What would settle it

Rerun the flag algebra calculation behind Proposition 3.3 and check that the $6$-vertex SDP really proves the claimed inequality between bad and missing $S_2$ copies under the stated constraints; alternatively, exhibit any $\{K_4^{3-},C_5^{3-}\}$-free $3$-graph family with $S_2$-density exceeding $6/13+o(1)$.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the $\ell_2$-norm Tur\'an problem for $C_\ell^{3-}$ collapses onto a single, recursively self-similar extremal family. The equivalent object studied is the density of $S_2$, the $3$-graph on four vertices with exactly two edges, in $\{K_4^{3-},C_5^{3-}\}$-free $3$-graphs: the maximum $S_2$-density is exactly $6/13$, which corresponds to $\ell_2$-norm density $1/26$. Near the maximum, a locally maximal $3$-partition exists in which the number of $S_2$ copies that involve a bad edge across parts is at most $9/10$ of the number of $S_2$ copies present in the complete $3$-partite graph but missing from $H$, up to $o(n^4)$; this comparison makes the recursive argument close. Iterating over the three parts produces both the upper bound and the stability statement: the extremal hypergraphs are exactly the Trec-subconstructions, up to $o(n^3)$ edge deletions.

Load-bearing premise

The argument depends on two computer-assisted bounds, one comparing bad and missing edges with constant $3/4$ and one comparing bad and missing $S_2$ copies with constant $9/10$, whose proofs are not fully present in the paper; if either bound is wrong, the main theorem has no support.

Editorial extensions

If this is right

  • For every integer $\ell\ge 5$ with $\ell\not\equiv 0\pmod 3$, the $\ell_2$-norm Tur\'an density of $C_\ell^{3-}$ is exactly $1/26$.
  • Near-extremality in the codegree-squared norm forces structure: deleting $o(n^3)$ edges from a hypergraph with norm at least $(1/26-\delta)n^4$ leaves a Trec-subconstruction.
  • The equivalent generalized Tur\'an statement holds: the maximum density of the two-edge $3$-graph $S_2$ in $\{K_4^{3-},C_5^{3-}\}$-free $3$-graphs is $6/13$.
  • Since $1/26$ exceeds the $1/27$ norm of the balanced complete $3$-partite hypergraph, the extremal construction must be genuinely recursive rather than a single-level partition.
  • The same computer-assisted framework also yields the exact value $\pi_{\ell_2}(F_{3,2})=1/8$ for a different $3$-graph whose ordinary Tur\'an density was previously known.

Reading between the lines

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

  • The reduction from arbitrary $\ell$ to the two fixed forbidden hypergraphs $K_4^{3-}$ and $C_5^{3-}$ is homomorphism-based, so the same $6/13$ bound might transfer to other families that degenerate to $C_5^{3-}$, provided the two fixed-hypergraph bounds continue to hold.
  • The stability theorem is strong enough to suggest that the exact $\ell_2$-norm extremal number $\mathrm{ex}_{\ell_2}(n,C_\ell^{3-})$ may be determined for large $n$ if the Trec construction satisfies a vertex-extendability condition of the kind used in other generalized Tur\'an problems; the paper does not establish this.
  • The main theorem currently rests on a long computer calculation whose certificate is stored outside the paper, so independent verification of that calculation, or a human-readable proof of the $S_2$-comparison, would make the $1/26$ value fully reproducible.
  • The inequality $1/26>1/27$ shows that the recursive iteration is not cosmetic: in the $\ell_2$ norm, the best single-level $3$-partition is strictly suboptimal, and the gain comes from reusing the same $1/26$ factor inside the parts.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the ℓ2-norm Turán density of the 3-uniform tight cycle minus one edge, C_ℓ^{3-}, for ℓ ≥ 5 with ℓ not divisible by 3. The main theorem, Theorem 1.1, asserts that π_{ℓ2}(C_ℓ^{3-}) = 1/26, together with an Erdős–Simonovits-type stability statement: every n-vertex C_ℓ^{3-}-free 3-graph with ℓ2-norm close to the extremal value is, after removing o(n^3) edges, a subgraph of the recursive three-partite Trec construction. The proof first reduces the ℓ2-norm problem to an S2-density problem for the family {K_4^{3-}, C_5^{3-}} via equation (1) and homomorphism arguments. Theorem 2.1, the core density and stability statement, is then proved by flag algebra: Propositions 3.1–3.3 provide a partition with a local maximality condition, a comparison of bad and missing edges with constant 3/4, and a comparison of bad and missing S2 copies with constant 9/10. Lemma 4.1 turns these into a recursive inequality, Fact 2.2 yields the density bound 6/13 for S2, and Section 5 turns the same ingredients into a stability proof. The lower bound comes from the independent Trec construction, whose ℓ2-norm is asymptotically n^4/26.

Significance. If the result is correct, it resolves the Balogh–Clemen–Lidický conjecture on π_{ℓ2}(C_5^{3-}) and extends it to all ℓ not divisible by 3, adding a strong stability theorem in the ℓ2-norm. The reduction from ∥·∥2 to S2-density is clean and elementary, and the recursive inequality in Lemma 4.1 is elegant. The paper also demonstrates that flag algebra methods can handle generalized Turán densities and stability in the ℓ2 setting, which is a valuable methodological contribution. However, the central proof depends on three computer-generated or externally cited propositions, and two of those (Propositions 3.1 and 3.2) are stated without proof, while the third (Proposition 3.3) relies on a 19-hour SDP whose certificate is only placed on an external Drive link. These gaps prevent the proof from being independently verifiable in its current form, and they are load-bearing rather than cosmetic.

major comments (3)
  1. [Section 3, Proposition 3.2] Proposition 3.2 is stated without proof, with the text saying only that 'the proofs are the same' as in [BLLP24, Proposition 3.3]. This is a load-bearing omission: the constant 3/4 enters the proof of Proposition 3.3 through the limit constraint (9), and if the true optimal constant were larger, the feasibility problem solved by the SDP would be different and the contradiction to (14) would not follow. The strengthened constant 0.75 is not a routine restatement of the earlier 0.99 result; the manuscript explicitly says the constant is crucial. The same concern applies to Proposition 3.1, whose proof is also omitted. The authors should either include full proofs adapted to the new hypotheses (S2-density assumption, local maximality, and the threshold 0.198) or provide a machine-checkable certificate for the strengthened inequality.
  2. [Section 3, Proposition 3.3] The proof of Proposition 3.3 is a computer-assisted flag algebra calculation with |F_0^6| = 16181 constraints and a reported runtime of about 19 hours, but the certificate is available only at an external Google Drive link. For a computer-assisted proof, the sum-of-squares decomposition or an equivalent certificate must be permanently archived with the paper or in a stable repository, so that the implication from constraints (8)–(13) to the non-positivity of the expression in (14) can be checked independently. As written, the main theorem relies on an unverifiable external artifact.
  3. [Section 5, proof of Lemma 5.1] In the last paragraph of the proof of Lemma 5.1, the text states: 'It follows from (21) and Proposition 3.1 that µH(V1,V2,V3) ≥ 0.918.' This is impossible, since the max-cut ratio µ is at most 2/9 ≈ 0.222 for any 3-graph. The intended value is almost certainly the lower bound 0.198 provided by Proposition 3.1, which is exactly what is needed to apply Proposition 3.2. The displayed constant 0.918 must be corrected; as written, this step is false and obscures the logical dependence of the stability proof on Proposition 3.2.
minor comments (4)
  1. [Section 3, Figures 1 and 2] The proof of Proposition 3.3 refers to Figure 1 and Figure 2 to catalogue the types of missing and bad S2 copies, but these figures are not included in the arXiv text. The classification is used in the definitions of Sb and Sm in (14), so the figures should be included in the paper.
  2. [Section 3, Proposition 3.3 proof] The sentence 'The results returned by the computer for the calculations for φ(Sb) − (9/10)φ(Sm) is indeed 0' is imprecise: the SDP presumably shows that the left-hand side of (14) is at most 0 under the constraints, not that it equals 0 as a value. Please state the exact output and the rounding/feasibility tolerances used.
  3. [Section 6, Theorem 6.2] The ancillary result on π_{ℓ2}(F_{3,2}) is presented with a certificate only on another external Drive link. If this result is part of the paper, it should also be accompanied by a permanent, verifiable certificate or by a proof; otherwise it should be moved to a clearly separate remark.
  4. [References] The paper cites [BLLP24] and [LMP24] as arXiv preprints. If either has been accepted or published in the meantime, the references should be updated.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the main bound is derived from independent flag-algebra constraints and a standalone calculus inequality; self-citation to [BLLP24] is a verification concern, not a circular reduction.

full rationale

The derivation chain is not circular. Theorem 1.1 is first reduced by a parameter-free identity, N(S2, H) = (||H||2 - 3|H|)/2 and pi(S2, F) = 12*pi_{\ell 2}(F), to the independent generalized Turan statement Theorem 2.1. The upper bound 6/13 is obtained from a recursive inequality in Lemma 4.1 together with Fact 2.2, whose constants do not encode the target value: Fact 2.2(i) is a standalone calculus inequality, and the Trec construction supplies the matching lower bound independently. The flag-algebra Proposition 3.3 is run under constraints (8)-(13) that do not include the target density 6/13 as an input; the target enters only as the assumed near-extremal threshold (13), which is a legitimate hypothesis for a stability and density proof. The main self-citation concern is Proposition 3.2, whose |B|-(3/4)|M| <= 0 bound is delegated to [BLLP24] with the note "Since the proofs are the same, we omit it here." This is an omitted or unverified proof and a load-bearing assumption, but it is not circular: the inequality is a lemma about {K4^{3-}, C5^{3-}}-free hypergraphs with a large max-cut ratio, and it neither assumes nor is definitionally equivalent to the S2-density upper bound 6/13. Likewise Proposition 3.1 is delegated to [BLLP24] but is not a restatement of the conclusion. The external Drive certificates for the 19-hour SDP are cited as evidence; failure to include or verify them is a reproducibility risk, not circularity. Overall, the paper's prediction is not forced by fitting parameters or by self-referential definitions; the only reason not to score 0 is the heavy reliance on the authors' own prior work for two propositions, which remains a verification concern rather than a circular reduction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The core proof is a chain of standard reductions plus a computer-assisted flag algebra step. The only 'free' items are external theorems and the unflagged assumption that the omitted proofs transfer. No empirical fitting parameters appear.

assumptions (5)
  • domain assumption The flag algebra method is sound and the SDP certificates supplied externally are valid.
    Used for Proposition 3.3; the paper reports the computer output but does not verify the certificates in the text.
  • ad hoc to paper Propositions 3.1 and 3.2 from [BLLP24] remain valid under the changed hypotheses (S2-density instead of edge density, µ ≥ 0.198, constant 0.75).
    The authors state 'the proofs are essentially the same' and omit them; this is an unproved load-bearing assumption.
  • domain assumption The Turán density of {K4^{3-}, C5^{3-}} equals 1/4 (Theorem 2.3 from [LMP24, BLLP24]).
    Used to bound edge density in the flag algebra setup and in the reduction from C_ℓ^{3-}-free graphs.
  • domain assumption For every ℓ ≥ 5 not divisible by 3 there is a homomorphism from C_ℓ^{3-} to C5^{3-} ([BL24, Claim 5.14]).
    Used in Section 2 to reduce the general ℓ case to C5^{3-}.
  • standard math The hypergraph removal lemma and supersaturation method are applicable.
    Used to pass from C_ℓ^{3-}-free to {K4^{3-},C5^{3-}}-free with negligible loss of S2 copies.

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Cite this review

Pith. "Pith review of Tur\'{a}n density of tight cycles minus one edge in the $\ell_2$-norm." pith.science (2026). https://pith.science/paper/33NANWEC

@misc{pith2026250700812,
  author       = {Pith},
  title        = {Pith review of: Tur\'an density of tight cycles minus one edge in the $\ell_2$-norm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/33NANWEC}},
  note         = {Machine review of arXiv:2507.00812}
}
abstract

The $3$-uniform tight $\ell$-cycle minus one edge $C_{\ell}^{3-}$ is the $3$-graph on $\ell$ vertices consisting of $\ell-1$ consecutive triples in the cyclic order. We show that for every integer $\ell \ge 5$ satisfying $\ell\not\equiv 0\pmod3$, every $C_{\ell}^{3-}$-free $3$-graph whose $\ell_2$-norm, that is, the sum of codegree squares, is close to the maximum must be structurally close to the iterative blowup of a single triple. This confirms a conjecture of Balogh--Clemen--Lidick\'{y}~[Surveys in combinatorics 2022, 21-63] in a stronger form.

Figures

Figures reproduced from arXiv: 2507.00812 by the authors.

Figure 1
Figure 1. Two types of missing S2 (there may be other edges not indicated in the figure). 7 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Six types of bad S2. The following result is an analogue of Proposition 3.2, but instead of comparing bad edges and missing triples, we compare bad and missing S2’s. Proposition 3.3. Suppose that H is an n-vertex {K3− 4 , C3− 5 }-free 3-graph with ρ(S2, H) ≥ 6/13−10−6 and V1∪V2∪V3 = V (H) is a locally maximal partition satisfying µH(V1, V2, V3) ≥ 0.198. Then with BS2 = BS H 2 (V1, V2, V3) and MS2 = MS H 2 (V1, V2, V… view at source ↗

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Reference graph

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