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REVIEW 4 major objections 3 minor 22 references

Exact Tur\'{a}n densities in triple systems

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Pith's one-line read The paper proves that the Turán density of the pair consisting of the tight 4-cycle $C_4^3$ and the complement of the five-vertex 3-graph $F_5$ with edges $\{123,145,245\}$ is exactly $2\sqrt{3}-3$, confirming a conjecture from [21]…

desk verdict Real new densities and a confirmed conjecture, but the proof outsources its core to an unpublished preprint and the notation is sloppy enough that the main theorem's statement is ambiguous. read the letter →

arxiv 2507.07360 v1 pith:CP3LIK4M submitted 2025-07-10 math.CO

Nannan Chen , Yuzhen Qi , Caihong Yang , Hongbin Zhao This is my paper
classification math.CO MSC 05C6505D0505C35
keywords Turándensity3-uniformhypergraphstightcyclesnon-principalfamiliesflagalgebrasemi-bipartite3-graphsextremalcombinatoricsShiconjecture
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

This paper proves exact Turán densities for three families of 3-uniform hypergraphs. Its main result is $\pi(C_4^3, \overline{F_5})=2\sqrt{3}-3$, where $C_4^3$ is the tight 4-cycle and $\overline{F_5}$ is the complement of the five-vertex 3-graph with edges $\{123,145,245\}$; this confirms the conjecture made in [21]. The other two results, $\pi(F_{3,2}, C_5^{3-})=2/9$ and $\pi(F_{3,2}, \text{induced complement of }F_{3,2})=3/8$, are strictly smaller than the individual Turán densities of the forbidden graphs, giving new instances of non-principality for triple systems. Exact densities for pairs of 3-graphs are rare, and the proofs transfer a computer-assisted flag algebra method to a new family with a structural extremal characterization.

What carries the argument

Three interlocking pieces carry the proof: the flag algebra method supplies three computer certificates that give a coarse upper bound on the density, a lower bound on the max-cut ratio of high minimum degree instances, and a relation between bad and missing edges in a locally maximal partition; Lemma 2.2 converts these certificates into a structural statement about a dense extremal example having a bipartition whose edge number is bounded by the semi-bipartite count plus internal edges of one part, up to a small error and a penalty $\max\{|B|/3999, |M|/4000\}$; and Fact 2.1, two one-variable quadratic inequalities, finishes the optimization, giving $2\sqrt{3}-3$ exactly. The named object throughout is the Brec-construction, the recursively built complete semi-bipartite 3-graph whose edge density approaches $2\sqrt{3}-3$ and which supplies the matching lower bound.

What would settle it

Run the semidefinite program for the $\{C_4^3,\overline{F_5}\}$-free family directly with the certificates stated in Section 3 and check whether the identity $u-f(H)=\mathrm{SOS}+\sum c_F p(F,H)+o(1)$ holds with $u=2\sqrt{3}-3$; the theorem is false if the optimal value exceeds $2\sqrt{3}-3$, or if a locally maximal partition of a dense $\{C_4^3,\overline{F_5}\}$-free 3-graph violates the inequality $|B|-\frac{3999}{4000}|M|\le \xi n^3$ for some $\xi>0$.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Turán density of the pair consisting of the tight 4-cycle $C_4^3$ and the complement $\overline{F_5}$ is exactly $2\sqrt{3}-3$, matching the recursive semi-bipartite construction and settling the conjecture from [21]. The mechanism is a structural lemma stating that every sufficiently dense $\{C_4^3,\overline{F_5}\}$-free 3-graph admits a partition $V_1\cup V_2$ with $n/2 \le |V_1| \le 4n/5$ whose edge count is at most $\binom{|V_1|}{2}|V_2|+|H[V_2]|+\xi n^3$ minus a penalty proportional to the number of bad and missing edges across the partition; from this, elementary quadratic inequalities in the part ratios force the density below $2\sqrt{3}-3$. The same approach yields the two further exact values $\pi(F_{3,2}, C_5^{3-})=2/9$ and $\pi(F_{3,2}, \text{induced complement of }F_{3,2})=3/8$, both strictly below the densities of their individual members.

Load-bearing premise

The argument's load-bearing premise is that Propositions 3.1, 3.2, and 3.3, flag-algebra bounds imported from a recent preprint and said there to have 'basically the same' proofs, are correct and apply verbatim to the family $\{C_4^3,\overline{F_5}\}$; if any of these three statements fails for this family, the upper bound of Theorem 1.1 collapses, and the current paper does not reproduce or verify those proofs.

Editorial extensions

If this is right

  • For every $\{C_4^3,\overline{F_5}\}$-free 3-graph on $n$ vertices, the maximum number of edges is $(2\sqrt{3}-3+o(1))\binom{n}{3}$, so the extremal density is now known exactly rather than only approximated.
  • Extremal examples are asymptotically captured by recursively semi-bipartite constructions, meaning any near-extremal family must have a large part whose internal structure is again of the same recursive type.
  • The pair $(F_{3,2}, C_5^{3-})$ is non-principal: forbidding both graphs yields density $2/9$, strictly less than the smaller of the two individual densities.
  • The pair $(F_{3,2}, \text{induced complement of }F_{3,2})$ has density $3/8$, again strictly below $\pi(F_{3,2})=4/9$, adding a new example to the short list of non-principal pairs of 3-graphs.
  • The method shows that flag-algebra upper bounds from [7] can be transferred to the $\{C_4^3,\overline{F_5}\}$ family, yielding an exact result rather than only an approximation.

Reading between the lines

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

  • The printed body consistently omits the complement bar, writing $\pi(C_4^3,F_5)$ and $\pi(F_{3,2},\text{induced }F_{3,2})$; this extraction follows the abstract's version, which states the second forbidden objects as complements. A reader using the body's literal notation would get statements that appear inconsistent with the cited lower bounds, so the body should be read as containing a notation
  • Because the lower bound for Theorem 1.1 comes from Brec-constructions, which avoid both $F_5$ and $\overline{F_5}$, the complement convention does not affect the lower bound; only the upper-bound proof's imported propositions depend on which family is forbidden.
  • The same scheme, a structural lemma plus imported flag-algebra certificates, is likely to yield exact densities for other pairs of short tight cycles and 5-vertex obstructions, once the three propositions are regenerated for the new family.
  • The strict inequalities in Theorems 5.2 and 5.3 hint that non-principality may be more common among pairs involving a fixed 5-vertex 3-graph than the original question in [18] suggests.
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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

4 major / 3 minor

Summary. The paper claims three exact Turán density results for 3-uniform hypergraphs. The headline result, Theorem 1.1, is stated as π(C_4^3, F_5)=2√3−3 with F_5 defined on [5] by edges {123,145,245}; the abstract instead says the second forbidden graph is the complement of F_5. The proof of the upper bound passes through a structural lemma, Lemma 2.2, whose proof depends on three flag-algebra propositions (Propositions 3.1–3.3) stated without proof and described as 'almost identical' to results in the unpublished preprint [7]. A lower bound is obtained from Brec constructions. The paper also states two non-principal-family results, Theorems 5.2 and 5.3, whose upper bounds are asserted to follow from flag algebra calculations without presenting certificates or details.

Significance. If the intended theorem is π(C_4^3, complement of F_5)=2√3−3, the numerical value is a known constant, but its confirmation for a new pair of forbidden 3-graphs would settle Shi's conjecture and would be of interest to the extremal hypergraph community. The paper provides ancillary code and certificates via GitHub, which is commendable and facilitates verification. However, the significance is currently contingent: the body and abstract disagree on the forbidden family, the core upper-bound machinery is outsourced to an unpublished preprint, and the two additional theorems are supported only by references to unshown computer calculations. As written, the contribution cannot be fully assessed.

major comments (4)
  1. [Section 1, Theorem 1.1, and Section 2, lower bound] The forbidden family is written inconsistently. The abstract states "complement of F_5", but Theorem 1.1, Lemma 2.2, and the lower-bound discussion all write {C_4^3, F_5} without the overline, with F_5 defined by E(F_5)={123,145,245}. With that definition, the sentence "Since C_5^3⊆F_5" is false, and the claim that every Brec-construction is {C_4^3,F_5}-free is also false: in a semi-bipartite blow-up of size at least 3, taking a,b∈V_1, c∈V_2, d∈V_1, e∈V_2 realizes the three edges 123, 145, and 245. Thus the theorem as printed is not the one whose proof is given. If the complement is intended, the overline must appear consistently, and the paper should state that C_5^3⊆overline{F_5}, which would make the upper bound an immediate corollary of [7] and clarify the role of Lemma 2.2.
  2. [Section 3, Proposition 3.3] Inequality (4), |B|−(3999/4000)|M|≤ξn^3, is the only source of the penalty term max{|B|/3999, |M|/4000} in Lemma 2.2, and therefore is load-bearing for Theorem 1.1. Yet Proposition 3.3 is stated for a {K_4^3, F_5}-free graph (with K_4^3 rather than C_4^3), and its proof is omitted with the note that it is "almost identical" to [7, Proposition 3.3]. The preprint [7] concerns {C_4^3, C_5^3}-free graphs, not the family in this paper unless the complement notation is resolved. The exact input family used in the SDP, and the transferability of the certificate from [7] to the current family, must be specified and verified; otherwise the upper bound is unsupported.
  3. [Section 4, proof of Lemma 2.2] The proof applies Proposition 3.2 to the subgraph G to obtain inequality (7), the large max-cut lower bound. Proposition 3.2 is also imported from [7] with no proof. If the imported results are valid only for {C_4^3,C_5^3}-free graphs, then their validity for the family in this paper depends on the containment between the forbidden families, which is exactly the issue left unresolved by the missing overlines. The paper needs either complete proofs of Propositions 3.1–3.3 or a precise, justified statement of why the proofs in [7] carry over verbatim to the family treated here.
  4. [Section 5, Theorems 5.2 and 5.3] Both upper bounds are asserted to "follow from a flag algebra calculation", but no certificates, scripts, or even a description of the SDP inputs are included for these theorems. Section 3 describes only the flag-algebra framework for Theorem 1.1, and the cited repositories are not tied to the specific computations for Theorems 5.2 and 5.3. As a result, the non-principality claims in these theorems cannot be checked from the manuscript.
minor comments (3)
  1. [Throughout] The manuscript contains several typographical slips: "conjected" should be "conjectured", "Tu´an" should be "Turán", "Pikurkho" should be "Pikhurko", and "for for" appears in the acknowledgments. These should be corrected.
  2. [Section 1] The sentence "Since C_5^3⊆F_5, it follows from the result in [7] that π(C_4^3,F_5)≥2√3−3" has the wrong inequality direction for the stated containment; if C_5^3 is a subgraph of the forbidden graph, the density of the larger forbidden family is at most that of {C_4^3,C_5^3}, not at least. The direction must be corrected once the intended family is fixed.
  3. [Section 3, Proposition 3.3] Proposition 3.3 is stated with "K_4^3" although all surrounding statements concern the tight 4-cycle C_4^3; this appears to be a typo, but it is consequential because the flag-algebra input family determines whether the inequality is applicable.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof imports flag-algebra lemmas from the external preprint [7], but the target Turán density is not reduced to itself or to a fitted parameter.

full rationale

The paper's derivation does not contain a circular reduction. The lower bound for Theorem 1.1 is given by explicit Brec constructions, which are verified to avoid the forbidden family. The upper bound follows from Lemma 2.2, whose proof invokes Propositions 3.1–3.3. These propositions are imported from the external preprint [7] with the note 'the proofs are basically the same, we omit them here', but that is a reliance on external support and an omitted-proof completeness concern, not a circularity: [7] does not prove the present theorem, the forbidden family is not identical to the one in [7], and the constants such as α3.2 and β3.2 are not fitted from the target value 2√3−3. Fact 2.1 is an elementary inequality, and the later algebra converts Lemma 2.2 into the claimed bound. No fitted parameter is renamed as a prediction, no load-bearing self-citation is used, and no uniqueness or ansatz is smuggled in via authors' own prior work. Thus no circular step can be exhibited under the required standard.

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

The central claim rests on a series of results imported from the unpublished preprint [7] and on unverified flag algebra certificates. No new entities are introduced, and the numeric constants (α3.1, α3.2, β3.2) are outputs of the SDP solver, not fitted to external data. The free parameter list is empty because the proof introduces no hand-picked constants that are later fit to data.

assumptions (3)
  • domain assumption The flag algebra SDP certificates produced by the provided scripts are correct and certify the stated inequalities.
    The proofs of Propositions 3.1-3.3 are omitted; the paper relies on computer-generated certificates that are not formally verified in the text (Section 3).
  • domain assumption The results of Bodnár, León, Liu and Pikhurko [7], including Propositions 3.1-3.3, Fact 2.1, Corollary 4.1 and Fact 4.2, are correct.
    These results are imported from a recent unpublished preprint and used without proof as the core machinery for Lemma 2.2 and Theorem 1.1.
  • standard math Known Turán densities π(C_5^{3-}) = 1/4 and π(F_{3,2}) = 4/9 are correct.
    Used directly in the proofs of Theorems 5.2 and 5.3, citing prior published work [16, 6, 13].

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Pith. "Pith review of Exact Tur\'{a}n densities in triple systems." pith.science (2026). https://pith.science/paper/CP3LIK4M

@misc{pith2026250707360,
  author       = {Pith},
  title        = {Pith review of: Exact Tur\'an densities in triple systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CP3LIK4M}},
  note         = {Machine review of arXiv:2507.07360}
}
abstract

In this paper, we prove several new Tur\'{a}n density results for $3$-graphs. We show: $\pi(C_4^3, \mathrm{complement\ of\ } F_5) = 2\sqrt{3} - 3$, $\pi(F_{3,2}, C_5^{3-}) = \frac{2}{9}$, and $\pi(F_{3,2}, \mathrm{induced\ complement\ of\ } F_{3,2}) = \frac{3}{8}$. The first result confirms the conjecture of Shi~[On Tur\'an denisties of small triple graphs, European J. Combin. 52 (2016) 95-102]. The other results give several special non-principal family posed by Mubayi and R\"odl~[On the Tur\'an number of triple systems, J. Combin. Theory A. 100 (2002) 135-152].

Figures

Figures reproduced from arXiv: 2507.07360 by the authors.

Figure 2.1
Figure 2.1. BH pV1, V2q and MH pV1, V2q. The dashed edge indicates that it belongs to H. The following fact is straightforward to verify. Here, S 2 :“ ␣ px1, x2q P R 2 : x1 ` x2 “ 1 and xi ě 0 for i P r2s ( is the standard 1-dimensional simplex. Fact 2.1 ([7]). The following inequalities hold for every px1, x2q P S 2 with x2 ă 1: (1) x 2 1 x2 2p1´x 3 2q ď 2 ? 3´3 6 . (2) Suppose that x1 P “ 1 2 , 1 ‰ . Then 1 2 x 2 1x2 ` 2 ? 3 … view at source ↗

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Reference graph

Works this paper leans on

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