The Tur\'{a}n density of short tight cycles

Jared Le\'on; Levente Bodn\'ar; Oleg Pikhurko; Xizhi Liu

REVIEW 3 major objections 4 minor 2 cited by

The Tur\'{a}n density of short tight cycles

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The asymptotic edge density of 3-graphs avoiding short tight cycles is exactly 2√3 − 3.

desk verdict Strong, honest paper that settles a batch of hypergraph Turán problems; the one concrete thing to check before publication is the sign transcription in Proposition 3.3. read the letter →

arxiv 2506.03223 v1 pith:7S5UYYEF submitted 2025-06-03 math.CO

Levente Bodnár , Jared León , Xizhi Liu , Oleg Pikhurko This is my paper

classification math.CO MSC 05C6505D0505C35

keywords Turándensitytightcycles3-uniformhypergraphsflagalgebramethodstabilityrecursiveconstructionsextremalhypergraphtheorycomputer-assistedproof

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 determines the Turán density of short 3-uniform tight cycles. For the pair {$C4^{3}$,$C5^{3}$} and for every single tight ℓ-cycle with ℓ ≥ 7 not divisible by 3, every C-free 3-graph on n vertices has at most (2√3−3+o(1)) binom(n,3) edges, and this is attained by a recursive two-part construction. It also proves an Erdős–Simonovits-type stability result: any near-extremal C-free 3-graph becomes a subconstruction of the recursive building after deleting o($n^{3}$) edges. For {$C4^{3}$,$C5^{3}$} the structure is exact enough to determine the maximum number of edges up to an additive constant.

What carries the argument

The Brec construction is the recursive two-part 3-graph: take a complete semi-bipartite layer B[V1,V2] whose edges contain exactly two vertices from V1, then repeat the construction inside V2; its optimal top-part fraction is γ = (3−√3)/2 and its edge density is α. The upper-bound machinery consists of two flag algebra SDP certificates on 6-vertex flags: Proposition 3.2 shows every dense C′-free graph has a cut with at least α3.2 $n^{3}$/6 transversal edges, and Proposition 3.3, applied to a locally maximal partition, bounds the number of bad edges by 3999/4000 times the number of missing edges plus o($n^{3}$). Lemma 4.1 converts these local comparisons into a global bound, and the stability proof iterates the comparison down the recursive levels.

What would settle it

Independently re-verify the SDP certificates behind Propositions 3.1–3.3 with a different solver, checking that the maximum of |B| − (3999/4000)|M| under local maximality is at most 0 for both 6-vertex flag sets; a violation, or an explicit C-free 3-graph with edge density above 2√3−3, would falsify Theorem 1.1.

Watch

Extended reading notes

Core claim

The central discovery is that all of these short tight cycles are governed by one constant, α = 2√3−3 ≈ 0.46410. Theorem 1.1 states that π(C) ≤ α for C equal to {$C4^{3}$,$C5^{3}$} or {$Cℓ^{3}$} with ℓ ≥ 7 and 3∤ℓ; the known Brec construction gives π(C) ≥ α, so the Turán density is exactly α. Theorem 1.2 adds stability: with n large, any C-free 3-graph with at least α binom(n,3) − $δn^{3}$ edges can be made a Brec subconstruction by removing at most $εn^{3}$ edges. Theorem 1.3 refines this for {$C4^{3}$,$C5^{3}$}: every extremal graph has a partition V1 ∪ V2 with |V1|/n close to γ = (3−√3)/2, and H \ H[V2] is exactly the complete semi-bipartite 3-graph B[V1,V2], yielding ex(n,{$C4^{3}$,$C5^{3}$}) = brec(n) + O(1). The proof passes from long cycles to a finite forbidden family {$C4^{3}$,F1,F2} by homomorphism and hypergraph removal, then proves the finite case with flag algebra certificates.

Load-bearing premise

The entire upper bound rests on computer-generated flag algebra certificates whose correctness is checked only by the authors' custom Python verifier, so an undetected bug in the certificates, the objective functions, or the verifier would break the density bound.

Editorial extensions

If this is right

For every ℓ ≥ 7 with ℓ not divisible by 3, the asymptotic density of Cℓ^3-free 3-graphs is exactly 2√3−3, independent of ℓ, with equality attained by Brec constructions.
Near-extremal C-free 3-graphs are structurally forced: after deleting o(n^3) edges they are subconstructions of the recursive Brec graphs, so the extremal family has no exotic dense alternatives.
For {C4^3,C5^3}, the maximum number of edges is known up to an additive constant at every order, and every maximum graph is a Brec construction with the optimal part ratio at every level except for small exceptional parts.
Because longer cycles reduce to the finite family {C4^3,F1,F2}, the same density constant applies simultaneously to all cycles in the family, and any one forbidden cycle forces the same asymptotic density.
The method does not settle the single tight 5-cycle C5^3: the corresponding SDP is considerably larger and, by the authors' numerical evidence, not strong enough to prove the needed local inequality.

Reading between the lines

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

Beyond the paper: the trick of first refining a partition to local maximality and then feeding those constraints into the SDP is transferable to other extremal problems with conjectured recursive or blow-up extremal constructions; it may close gaps that bare flag algebra calculations leave open.
Beyond the paper: the reduction to a finite homomorphism-invariant obstruction set suggests that all tight 3-cycles with ℓ ≡ 1 or 2 mod 3 form one density class; a natural test is whether other blowup-invariant families built from homomorphic images of a single cycle share the same constant.
Beyond the paper: because the upper bound rests on machine-verified certificates, an independent verification with a different SDP solver or a formal proof assistant would convert the computational step into a routine check, leaving the recursive-decomposition argument as the sole mathematical content.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

This is a genuine advance. It proves π(Cℓ^3)=2√3−3 for every ℓ≥7 not divisible by 3, not just for sufficiently large ℓ, and it also handles the pair {C4^3,C5^3}, with a stability theorem and an O(1) extremal count. The local-maximality refinement is a real idea, and the analytic skeleton—Proposition 3.3 feeding Lemma 4.1, then the recursive density argument—is clear and internally consistent. Fact 2.9 and the inequalities in Section 4 check out.

The paper is also honest about its computational dependence. It ships certificates and a self-contained Python verifier, and the code is structured so the objective functions are hard-coded and visible. That is more than most flag algebra papers do, and it should count in the paper's favor.

The main residual risk is exactly what the reader flagged: the SDP-to-certificate pipeline. The stress-test note pinpoints a concrete place to look. The text in the proof of Proposition 3.3 says, after assuming the negation, that ϕ(M) − (3999/4000)ϕ(B) ≥ 6ξ, where the inequality needs to be ϕ(B) − (3999/4000)ϕ(M) ≥ 6ξ. That is almost certainly a typo in the write-up—it contradicts equation (15) directly above it. But it is precisely the kind of sign reversal that a hard-coded objective in the verifier could inherit without anyone noticing. The authors should be asked to confirm, from the certificate itself, that the certified inequality matches (14). This is a five-minute check, and it would remove the one nagging doubt.

Other concerns are minor. The concluding remark that the C5^3-alone SDP is not strong enough is honest and useful. Section 6's graph lemma is dense but appears correct. The citation pattern looks appropriate; the overlap with the authors' own prior work [4] is methodological, not circular, since the target α comes from the independent Brec construction and is not fed into the SDPs.

This paper deserves a serious referee. The result is important within extremal combinatorics, the proof is detailed, and the computational artifacts are unusually well documented. I would send it out, with the specific request to verify the Proposition 3.3 sign and ideally to add one independent sanity check on the verifier's objective transcription.

Referee Report

3 major / 4 minor

Summary. The paper determines the Turán density of the family C = {C_4^3, C_5^3} and of every single tight cycle C_ℓ^3 with ℓ ≥ 7 and ℓ not divisible by 3, proving that in each case the Turán density equals α = 2√3 − 3. The lower bound comes from the known Mubayi–Rödl Brec construction, and the main work is an upper bound obtained through flag-algebra computations. The paper also proves an Erdős–Simonovits-type stability theorem and, for {C_4^3, C_5^3}, a finer structural result determining ex(n, {C_4^3, C_5^3}) up to an additive O(1) error.

Significance. If the result is correct, it resolves the Turán density of short tight cycles in the three-uniform case, extending the large-ℓ result of Kamčev–Letzter–Pokrovskiy to all ℓ ≥ 7 not divisible by 3. The analytic framework is strong: the reduction to the two auxiliary families C′ via blowup-invariance, the recursive local-maximality argument, and the clean use of Fact 2.9 are elegant and largely machine-checkable in their non-computational parts. The paper also ships its flag-algebra certificates and a self-contained Python verifier, which is a genuine strength for reproducibility. The main caveat, appropriately acknowledged by the authors, is that the decisive Proposition 3.3 rests on non-human-checkable certificates verified by custom code.

major comments (3)

[§3, Proposition 3.3] There is a mismatch between the statement of Proposition 3.3 and its proof. The proof begins by assuming the proposition fails and then asserts the existence of a sequence satisfying, among other things, item (i): |H_n| ≥ β_{3.2} v(H_n)^3/6. The negation of the stated proposition does not yield this edge-density lower bound; it yields only the two bullet assumptions of the proposition together with the failure of (14). The lower bound (i) is nevertheless used to derive (18), so it is essential to the flag-algebra calculation. Either Proposition 3.3 must be restated with the additional hypothesis |H| ≥ β_{3.2} n^3/6, or the proof must deduce that hypothesis from the stated assumptions. In Lemma 4.1 the graph G to which Proposition 3.3 is applied does satisfy this stronger bound, so the application can be repaired, but the proposition as stated is not what is proved.
[§3, proof of Proposition 3.3, final paragraph] The final contradiction reverses the inequality in (15). From (15), the limit satisfies φ(B) − (3999/4000)φ(M) ≥ 6ξ, not φ(M) − (3999/4000)φ(B) ≥ 6ξ. As written, the sentence does not contradict the stated SDP conclusion that |B| − (3999/4000)|M| is at most 0. I assume this is a typo, but it is precisely the kind of sign/transcription error that can be inherited by a verifier with hard-coded objective functions. Please correct the inequality and explicitly confirm, by reference to the shipped verifier output, that the certificates certify the inequality |B| − (3999/4000)|M| ≤ 0 and not its reverse.
[§3, computer verification] Because Theorem 1.1 depends on Proposition 3.3, whose correctness is established only by the custom Python verifier with hard-coded objective functions, and because none of the certificates is human-checkable, I ask that the authors add to the verifier an explicit assertion, written in the notation of (12)–(14), of the exact rational objective being bounded for each SDP. A small independent sanity check — for example, evaluating the declared objective on a few random flags and comparing it with a direct computation of |B| − (3999/4000)|M| — would address the transcription risk directly and make the computational part of the proof more robust.

minor comments (4)

[§3] The labels “F act 2.1” and “F act 2.2” contain a typo and should read “Fact 2.1” and “Fact 2.2”.
[§6, proof of Claim 6.6] The proof of Claim 6.6 ends with “This completes the proof of Claim 6.4,” which is the wrong claim label.
[§6, proof of Theorem 1.3] The assertion that H′ = (H ∪ M) \ B is still {C_4^3, C_5^3}-free is stated without proof; a one-sentence explanation of why adding the missing semi-bipartite triples and deleting the bad triples cannot create a new tight 4- or 5-cycle would improve readability.
[§4, proof of Lemma 4.1] The phrase “we obtain through calculations on computer with rational numbers that 55/100 ≤ x_1 ≤ 78/100” would be easier to verify if accompanied by the explicit polynomial inequality x_1^2(1−x_1) ≥ α_{3.2}/3 and its solution interval, rather than referring to computer arithmetic.

Circularity Check

3 steps flagged · score 3.0 of 10

Paper is largely self-contained with an independent target constant, but it relies on its own prior arXiv preprint for a load-bearing stability tool and on a custom non-human-checkable verifier; no constructional circularity found.

self citation load bearing [Section 1, Proof outline; Section 3, first paragraph]
"Our proofs of the corresponding results for C' crucially use the flag algebra machinery developed by Razborov [22] and are computer-assisted. More specifically, we adopt the strategy used in the previous work [4] for determining the Turán density of C_5^{3-} ..."

Theorem 1.1's proof relies on the strategy, including the manual local-maximality refinement of the SDP, imported from the authors' own paper [4], which is an unreviewed arXiv preprint at the time of writing. The load-bearing methodological premise (that this refined flag algebra setup suffices) is justified by reference to the authors' own prior work, not by independent verification; a failure of that strategy in [4] would undermine the present proof. This is a partial self-citation that is load-bearing, though not definitional.
fitted input called prediction [Section 3, Proposition 3.2 proof, final paragraph]
"(In fact, our definition of α_3.2 comes from taking the rounded lower bound on ϕ([[Y_E]]) returned by the computer and dividing it by α_3.1.)"

The constant α_3.2 in Proposition 3.2 is not an independently predicted threshold; it is defined as the rounded SDP-computed lower bound for ϕ([[Y_E]]) divided by α_3.1, and β_3.2 is then chosen (4641/10000) just below the target α. The proposition's conclusion µ(H) ≥ α_3.2 is therefore true by the way the constants were constructed from the same computation it claims to certify. This is a parameter chosen to make the proof work rather than a prediction, a mild form of fitted-input circularity.

1 more flagged steps

self definitional [Section 3, final paragraph of Proposition 3.3 proof]
"The results returned by the computer for upper bounding |B| − 3999/4000 |M| are indeed 0 in both cases ... However, by the assumption in (15), φ satisfies φ(M) − 3999/4000 φ(B) ≥ 6ξ > 0, a contradiction."

The displayed contradiction is not a contradiction with φ(M) − (3999/4000)φ(B) ≥ 6ξ, since this inequality is merely the negation of φ(B) − (3999/4000)φ(M) ≤ 0 only if the roles of B and M were swapped. The text has φ(M) − (3999/4000)φ(B) ≥ 6ξ, i.e. the reverse of the objective φ(B) − (3999/4000)φ(M) that was certified to be ≤ 0. As written, the two inequalities can both hold, so the alleged contradiction is self-inconsistent rather than a valid reduction. This is likely a typo, but the line bears exact Eq. (15) and the final contradiction.

full rationale

The central quantity α = 2√3−3 is determined by the Mubayi–Rödl Brec construction and is not fed into the flag algebra SDPs, which instead prove independent numerical bounds (α_3.1, α_3.2, and the 3999/4000 inequality). The final deduction in Theorem 2.4 compares these bounds through Fact 2.9, so there is no equation-level construction in which the target density is an input. The paper is self-contained against external benchmarks in the sense that the Brec construction gives a genuine lower bound and the SDP certificates are verified by a separate Python program, albeit not human-checkable. However, the proof outline in Section 1 explicitly adopts the strategy of the authors' own unpublished [4] for the tight 5-cycle minus an edge, including the method of adding local maximality manually into the SDP; the stability Theorem 1.3 and the recursive step of Theorem 2.5 also invoke Lemma 5.1, and Section 7 shows the same SDP approach is known to fail for C_5^3, so the success of this strategy for the present families is a nontrivial methodological assumption sourced from the same group's prior work. Separately, the constants α_3.2 and β_3.2 are defined by rounding the computer's own returned lower bound, and the final line of Proposition 3.3's proof contains an apparent swap of B and M in the displayed contradiction; if that swap reflects the actual certificate then Proposition 3.3 and Theorem 1.1 would not follow. For all these reasons I cannot give a 0; the paper has no constructional circularity, but its reliance on self-citation for a load-bearing method and the fitted-constant/verifier gaps warrant a moderate score of 3.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

No data-fitted parameters appear; the target α=2√3−3 comes from the independent Brec construction, and the flag algebra SDP computes bounds rather than fitting α. The three listed entries are hand-chosen proof constants, not fitted quantities. The proof also depends on standard background theorems and on the correctness of the supplied verification software.

free parameters (3)

coefficient 3999/4000 in Proposition 3.3 = 3999/4000
Hand-chosen constant strictly less than 1; it converts the SDP bound |B|−(3999/4000)|M|≤o(n^3) into the slack needed for the recursive density argument in Lemma 4.1.
β3.2 = 4641/10000 = 4641/10000
Hand-chosen lower bound on edge density in Proposition 3.2, set just below α−10^−6 to make the SDP premise non-vacuous.
β3.3 = 2/5 = 2/5
Hand-chosen lower bound on the max-cut density required by Proposition 3.3; large enough to force a nontrivial partition and small enough to be satisfied after the earlier proof steps.

assumptions (4)

standard math Flag algebra limit theory and positivity of densities (Razborov 2007)
Invoked in Section 3 to convert finite SDP certificates into asymptotic inequalities for all sufficiently large C'-free 3-graphs.
standard math Hypergraph Removal Lemma and Erdős's subgraph-count bound for H-free graphs
Used in Section 2 to justify passing from the forbidden family C to the smaller family C'={C4,F1,F2} or {C4,C5} with only o(n^3) edge loss.
domain assumption Blowup-invariance of the forbidden families C'
Used for the boundedness and smoothness results in Proposition 2.6; the paper says the property is routine to check, but the check is not fully written out.
ad hoc to paper Correctness of the custom Python verifier and SDP output files
Section 3 relies on this verifier because the certificates are not human-checkable; an independent audit of the verifier is not included in the paper.

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Pith. "Pith review of The Tur\'{a}n density of short tight cycles." pith.science (2026). https://pith.science/paper/7S5UYYEF

@misc{pith2026250603223,
  author       = {Pith},
  title        = {Pith review of: The Tur\'an density of short tight cycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7S5UYYEF}},
  note         = {Machine review of arXiv:2506.03223}
}

abstract

The $3$-uniform tight $\ell$-cycle $C_\ell^{3}$ is the $3$-graph on $\{1,\dots,\ell\}$ consisting of all $\ell$ consecutive triples in the cyclic order. Let $\mathcal{C}$ be either the pair $\{C_{4}^{3}, C_{5}^{3}\}$ or the single tight $\ell$-cycle $C_{\ell}^{3}$ for some $\ell\ge 7$ not divisible by $3$. We show that the Tur\'an density of $\mathcal{C}$, that is, the asymptotically maximal edge density of a large $\mathcal{C}$-free $3$-graph, is equal to $2\sqrt{3} - 3$. We also establish the corresponding Erd\H{o}s-Simonovits-type stability result, informally stating that all almost maximum $\mathcal{C}$-free graphs are close in the edit distance to a 2-part recursive construction. This extends the earlier analogous results of Kam\v{c}ev-Letzter-Pokrovskiy ["The Tur\'an density of tight cycles in three-uniform hypergraphs", Int. Math. Res. Not. 6 (2024), 4804-4841] that apply for sufficiently large $\ell$ only. Additionally, we prove a finer structural result that allows us to determine the maximum number of edges in a $\{C_{4}^{3}, C_{5}^{3}\}$-free $3$-graph with a given number of vertices up to an additive $O(1)$ error term.

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Forward citations

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

Bodn´ ar, J

L. Bodn´ ar, J. Le´ on, X. Liu, and O. Pikhurko,The Tur´ an density of the tight5-cycle minus one edge, arXiv preprint arXiv:2412.21011 (2024)

arXiv 2024

[2] [1]

Baber and J

R. Baber and J. Talbot,Hypergraphs do jump, Combin. Probab. Computing20(2011), 161–171

work page 2011

[3] [2]

Balogh, P

J. Balogh, P. Hu, B. Lidick´ y, and F. Pfender,Maximum density of induced5-cycle is achieved by an iterated blow-up of5-cycle, European J. Combin.52(2016), 47–58. MR3425964

work page 2016

[4] [3]

Balogh and H

J. Balogh and H. Luo,Tur´ an density of long tight cycle minus one hyperedge, Combinatorica44 (2024), no. 5, 949–976. MR4805884

work page 2024

[5] [5]

M. K. de Carli Silva, F. M. de Oliveira Filho, and C. M. Sato,Flag algebras: a first glance, Nieuw Arch. Wiskd. (5)17(2016), no. 3, 193–199. MR3643687

work page 2016

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Erd˝ os,On extremal problems of graphs and generalized graphs, Israel J

P. Erd˝ os,On extremal problems of graphs and generalized graphs, Israel J. Math.2(1964), 183–190. MR183654

work page 1964

[7] [7]

Erd˝ os,Some recent results on extremal problems in graph theory

P. Erd˝ os,Some recent results on extremal problems in graph theory. Results, Theory of Graphs (In- ternat. Sympos., Rome, 1966), 1967, pp. 117–123 (English); pp. 124–130 (French)

work page 1966

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Erd˝ os and M

P. Erd˝ os and M. Simonovits,A limit theorem in graph theory, Studia Sci. Math. Hungar.1(1966), 51–57. MR205876

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