Pith. sign in

REVIEW 6 minor 3 cited by

The Tur\'an density of the tight 5-cycle minus one edge

T0 review · 0 major / 6 minor · reviewed 2026-08-10 · deepseek-v4-flash

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

Pith's one-line read The paper proves the Turán density of the tight 5-cycle minus one edge is exactly 1/4.

desk verdict A careful, transparent resolution of a 2011 conjecture, with the main caveat being the un-auditable 180-hour computer proof. read the letter →

arxiv 2412.21011 v3 pith:MQPI4KGW submitted 2024-12-30 math.CO

Levente Bodnár , Jared León , Xizhi Liu , Oleg Pikhurko This is my paper · ORCID
classification math.CO MSC 05C6505C3505D40
keywords TurándensityhypergraphstightcycleminusoneedgeTrecconstructionflagalgebrassemidefiniteprogrammingcomputer-assistedproofextremalstability
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 exact Turán density of the tight 5-cycle minus one edge, a five-vertex 3-uniform pattern, and shows that it is exactly one quarter. This means that, asymptotically, the largest possible edge fraction in any 3-uniform hypergraph avoiding the pattern is $1/4$, and the same value is proved for every tight $\ell$-cycle minus one edge with $\ell \ge 5$ and $3 \nmid \ell$. Exact values of this kind are rare for 3-uniform hypergraphs, and the result confirms a 2011 conjecture while extending a recent result that covered only sufficiently large cycle lengths. The proof is computer-assisted but adds a new structural step, local refinement, which feeds local optimality of a vertex partition into the flag-algebra calculation.

What carries the argument

The central object is the Trec construction, a recursively built 3-uniform hypergraph obtained by splitting vertices into three nonempty parts, adding every triple that crosses all three parts, and recursing inside each part; it provides the lower bound and the conjectured extremal shape. The mechanism that carries the proof is a flag-algebra semidefinite program augmented by local refinement. The new combinatorial input is that for any vertex partition there exists a locally maximal partition with at least as many transversal edges, where locally maximal means no single vertex move increases the number of triples crossing all three parts. This condition is added to the SDP and yields four computer-verified propositions: a near-sharp upper bound on edge density, a lower bound on the max-cut ratio, an inequality $|B| \le \frac{99}{100}|M|$ comparing bad triples (two vertices in one part and one in another) with missing transversal triples, and a vertex-stability bound for link graphs. These propositions feed a recursion that bounds the size of any $C_5^{3-}$-free hypergraph by the size of the best Trec construction up to lower-order terms.

What would settle it

Run the published certificates through an independent exact-rational verifier and exhibit a violation of any of the four computer propositions, or find a $C_5^{3-}$-free 3-graph with edge density above $(1/4+\varepsilon)\binom{n}{3}$ for arbitrarily large $n$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is $\pi(C_5^{3-})=1/4$, where $C_5^{3-}$ is the 3-uniform hypergraph on five cyclically ordered vertices whose edges are four consecutive triples. The lower bound comes from the Trec construction: partition the vertices into three nonempty parts, take all triples that meet all three parts, and repeat recursively inside each part; these hypergraphs avoid $C_5^{3-}$ and have asymptotic density $1/4$. The upper bound is established by flag-algebra semidefinite programs combined with the observation that every vertex partition can be replaced by a locally maximal partition with at least as many transversal edges. The same route gives the density $1/4$ for every $C_\ell^{3-}$ with $\ell \ge 5$ and $3 \nmid \ell$, a stability theorem saying that near-extremal hypergraphs become Trec-subconstructions after deleting $o(n^3)$ edges, and an exact structure theorem: every maximum $C_5^{3-}$-free hypergraph on large $n$ is the complete 3-partite 3-graph on three nearly equal parts together with recursively extremal hypergraphs inside the parts.

Load-bearing premise

The upper-bound proof rests on four computer-generated flag-algebra propositions whose certificates are verified only by the authors' own Python verifier; if that verifier or one of the certificates contains an error, the proof of $\pi(C_5^{3-})=1/4$ collapses.

Editorial extensions

If this is right

  • For every integer $\ell \ge 5$ with $3 \nmid \ell$, $\pi(C_\ell^{3-})=1/4$.
  • Any $C_5^{3-}$-free 3-graph with density within $\delta n^3$ of the extremal value becomes a Trec-subconstruction after deleting at most $\varepsilon n^3$ edges.
  • Every maximum $C_5^{3-}$-free 3-graph on $n$ sufficiently large vertices splits into three nearly equal parts whose cross-edges are complete and whose internal parts are themselves maximum; hence $\mathrm{ex}(n,C_5^{3-})=\mathrm{trec}(n)+O(n)$.
  • The paper's boundedness and smoothness results make $C_5^{3-}$ available for tilting-type extremal applications, as noted in the concluding remarks.

Reading between the lines

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

  • Beyond the paper, the local-refinement device looks portable: any flag-algebra proof that bounds a cut-like quantity could first pass to a locally maximal partition and then encode that optimality as an SDP constraint, potentially pushing other near-miss Turán bounds to equality.
  • Beyond the paper, the same strategy should transfer to recursive extremal families built from tournaments, where the natural local move is reorienting one pair; the paper's concluding remarks point in this direction.
  • Beyond the paper, a quantitative stress test would be to recompute the four computer propositions without the added local-maximality constraints; the difference between the resulting bound and $1/4$ would measure exactly what the new idea contributes.
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

0 major / 6 minor

Summary. The paper determines the Turán density of the tight 5-cycle minus one edge, C_5^{3-}, proving that it equals 1/4. It extends this result to every tight ℓ-cycle minus one edge with ℓ ≥ 5 and ℓ not divisible by 3, and proves an Erdős–Simonovits-type stability theorem as well as a refined structural theorem for maximum C_5^{3-}-free 3-graphs. The upper-bound proof combines four computer-generated flag-algebra SDP propositions (Propositions 3.1–3.4) with a recursive 'local refinement' argument via Lemmas 4.1 and 5.1, Proposition 6.1, and the Hypergraph Removal Lemma.

Significance. This settles a 2011 conjecture of Mubayi–Pikhurko–Sudakov and extends Balogh–Luo's result from all sufficiently large ℓ to all ℓ ≥ 5 with ℓ not divisible by 3. The stability and exact-structure theorems go beyond the density value and provide strong structural information. A notable strength is that all computer-generated statements are accompanied by certificates and a self-contained Python verifier, making the machine-assisted part checkable in principle; the authors also report that their certificate size is far smaller than that of the independent work [19]. The main argument introduces a 'local refinement' step that is likely to be useful for other Turán-type problems.

minor comments (6)
  1. [Section 3 (Some remarks on our implementation)] The four computer-generated propositions are load-bearing for Theorem 1.1, and the paper states that the certificates are not human-checkable and that full verification takes about 180 hours. Please provide SHA-256 checksums of the verifier and all certificate files, together with the exact Python version and dependency list, so that the stated verification is reproducible in principle.
  2. [Section 4, proof of Lemma 4.1] The sentence 'Notice that it suffices to show that...' needs a short justification: the lemma's conclusion is stated for H with B_H and M_H, while the subsequent argument bounds the subgraph G obtained from the Removal Lemma. Please include the transfer argument using |H| - |G| ≤ ξn^3/3 and the relations between B_G, M_G and B_H, M_H.
  3. [Section 6, proof of Theorem 1.3] The assertion 'It is easy to see that G is C_5^{3-}-free' after deleting B and adding M is the key step that yields the exact structure; please expand it into a short proof, for example by showing that a C_5^{3-} in the complete 3-partite graph plus internal edges would force a bad edge.
  4. [Section 3, proof of Proposition 3.5] In the verification of the hypotheses of Proposition 3.4, the second 'Let us check Item (iii)' should read 'Item (iv)'.
  5. [Section 3, definition of μ(H)] In the definition of the max-cut ratio, 'V1, V2, V2 form a partition' should read 'V1, V2, V3'.
  6. [Section 5, proof of Theorem 1.2] There is a duplicated 'where where' in the choice of N4.1 and N5.1; the base case of the induction could also be stated more explicitly, though the argument is clear.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the upper bound is certified by independent flag-algebra propositions and the recursive argument; the self-citations are attribution or accompanied by proofs.

full rationale

The derivation of Theorem 1.1 uses Lemma 4.1, whose proof depends on Propositions 3.1–3.4 and Fact 2.1. Propositions 3.1–3.4 are computer-certified flag-algebra/SDP claims; their constants are outputs of rounding calculations, not fitted to the target: Section 3.1 states "The stated values for α3.1 and α3.2 are exactly the constants returned by our rounding calculations." The final inequality α ≤ 1/4 is obtained from Fact 2.1, an explicit analytic inequality, not from the SDP certificates. The lower bound is the Trec construction, which is defined and verified in the introduction by the recurrence trec(n) = max{n1n2n3 + trec(n1) + trec(n2) + trec(n3)}, so the citation [22] is attribution only. Theorem 1.2 follows from Lemma 5.1 and an induction; Theorem 1.3 uses Proposition 6.1, whose maximum-degree part is proven recursively and whose minimum-degree part cites the unpublished [20] but immediately supplies the deleting-duplicating proof for C_5^{3-}. Thus [20] is not load-bearing. The admitted limitation that the certificates are not human-checkable ("none of the certificates ... can be made sufficiently compact to be human-checkable") is a correctness/verification risk, not a circular step: no bound or equation in the proof is identical to the target by construction, and no fitted parameter is renamed as a prediction. No circular step was found.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The proof introduces no new mathematical entities. The listed free parameters are explicit proof constants tuned by computer experiment to make the SDP certificates exist; they are not empirical fits. The main unverified premise is the correctness of the machine-generated certificates and the authors' verifier.

free parameters (4)
  • β3.2 (edge-density threshold in Proposition 3.2) = 2499/10000 = 1/4 - 10^-5
    Chosen by experiment so that the SDP certificate for Proposition 3.2 exists; a slightly stronger value β3.2 = 1/4 - 10^-4 did not suffice (Section 3.1). This is a proof constant, not a fit to the final Turán density.
  • β3.3 (transversal-edge threshold in Proposition 3.3) = 19/100
    Chosen by experiment; β3.3 = 0.185 did not seem to suffice for the certificate (Section 3.1).
  • Ratio 99/100 in Proposition 3.3 = 99/100
    A fixed constant in the inequality |B| ≤ (99/100)|M|, output of the computer proof and part of the SDP design.
  • Ratio 9/10 in Propositions 3.4 and 3.5 = 9/10
    A fixed constant in the vertex-stability inequality; chosen to make the flag algebra proof work.
assumptions (5)
  • standard math Flag algebra framework of Razborov
    The paper uses the flag algebra calculus for limits of hypergraph sequences, described in Section 3 with references to [27, 28].
  • standard math Hypergraph Removal Lemma
    Invoked in the proof of Lemma 4.1 to pass to a {K_3^{4-}, C_5^{3-}}-free subgraph.
  • standard math Supersaturation Method
    Used in Section 2, equation (2), to reduce the Turán density of C_5^{3-} to that of the pair {C_5^{3-}, K_3^{4-}}.
  • ad hoc to paper Correctness of the computer-generated flag algebra certificates and the authors' Python verifier
    Propositions 3.1-3.4 are established by SDP certificates checked only by the provided verifier; the certificates are not human-checkable (Section 3).
  • domain assumption The weaker local-maximality inequality (10) in Proposition 3.3
    The flag algebra proof of Proposition 3.3 encodes only the averaged-random-move version of local maximality, losing the full condition (9); this is a modeling choice in the SDP.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Tur\'an density of the tight 5-cycle minus one edge." pith.science (2026). https://pith.science/paper/MQPI4KGW

@misc{pith2026241221011,
  author       = {Pith},
  title        = {Pith review of: The Tur\'an density of the tight 5-cycle minus one edge},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MQPI4KGW}},
  note         = {Machine review of arXiv:2412.21011}
}
abstract

Let the tight $\ell$-cycle minus one edge $C_\ell^{3-}$ be the $3$-graph on $\{1,\dots,\ell\}$ consisting of $\ell-1$ consecutive triples in the cyclic order. We show that, for every $\ell\ge 5$ not divisible by $3$, the Tur\'an density of $C_{\ell}^{3-}$ is $1/4$ and also prove some finer structure results. This proves a conjecture of Mubayi--Sudakov--Pikhurko from 2011 and extends the results of Balogh--Luo [Combinatorica 44 (2024) 949--976] who established analogous claims for all sufficiently large $\ell$. Results similar to ours were independently obtained by Lidick\'y--Mattes--Pfender [arXiv:2409.14257].

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

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

    math.CO 2025-07 conditional novelty 7.0 of 10

    The ℓ2-norm Turán density of the tight cycle minus one edge C_ℓ^{3-} is exactly 1/26 for every ℓ ≥ 5 with ℓ not divisible by 3, with a stability theorem.

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

    math.CO 2025-06 accept novelty 7.0 of 10

    The Turán density of every 3-uniform tight cycle of length ℓ≥7 with ℓ not divisible by 3, and of the pair {C4^3,C5^3}, is exactly 2√3−3.

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

    math.CO 2025-07 conditional novelty 6.0 of 10

    The authors prove exact Turán densities for three families of 3-graphs, including confirming Shi's conjecture that π(C_4^3, complement of F_5) equals 2*sqrt(3) - 3.

Reference graph

Works this paper leans on

32 extracted references · 29 canonical work pages · cited by 3 Pith papers

  1. [19]

    Lidicky, C

    B. Lidicky, C. Mattes, and F. Pfender, The hypergraph Tur´ an densities of tight cycles minus an edge, arXiv preprint arXiv:2409.14257 (2024)

    arXiv 2024

  2. [1]

    Baber and J

    R. Baber and J. Talbot, Hypergraphs do jump, Combin. Probab. Computing 20 (2011), 161–171

    work page 2011

  3. [2]

    Balogh, P

    J. Balogh, P. Hu, B. Lidick´ y, and F. Pfender, Maximum density of induced 5-cycle is achieved by an iterated blow-up of 5-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 , Combinatorica 44 (2024), 949–976

    work page 2024

  5. [4]

    Bodn´ ar, J

    L. Bodn´ ar, J. Le´ on, X. Liu, and O. Pikhurko,The Tur´ an density of short tight cycles, arXiv preprint arXiv:2506.03223 (2025)

    arXiv 2025

  6. [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), 193–199

    work page 2016

  7. [6]

    J. Deng, J. Hou, X. Liu, and C. Yang,Tight bounds for rainbow partial F -tiling in edge-colored complete hypergraphs, arXiv preprint arXiv:2406.14083 (2024)

    work page Pith review arXiv 2024

  8. [7]

    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

Show all 32 references
  1. [8]

    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)

    1966

  2. [9]

    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

    1966

  3. [10]

    Erd¨ os and A

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

    1946

  4. [11]

    Falgas-Ravry, O

    V. Falgas-Ravry, O. Pikhurko, E. Vaughan, and J. Volec, The codegree threshold of K − 4 , J. Lond. Math. Soc. (2) 107 (2023), no. 5, 1660–1691. MR4585299

    2023

  5. [12]

    Gilboa, R

    S. Gilboa, R. Glebov, D. Hefetz, N. Linial, and A. Morgenstern, On the local structure of oriented graphs—a case study in flag algebras , Electron. J. Combin. 29 (2022), Paper No. 3.39, 53

    2022

  6. [13]

    Glebov, D

    R. Glebov, D. Kr´ al’, and J. Volec, A problem of Erd˝ os and S´ os on 3-graphs, Israel J. Math. 211 (2016), no. 1, 349–366. MR3474967

    2016

  7. [14]

    W. T. Gowers, Hypergraph regularity and the multidimensional Szemer´ edi theorem, Ann. of Math. (2) 166 (2007), no. 3, 897–946. MR2373376

    2007

  8. [15]

    J. Hou, H. Li, X. Liu, L.-T. Yuan, and Y. Zhang, A step towards a general density Corr´ adi–Hajnal theorem, arXiv preprint arXiv:2302.09849 (2023)

    2023 arXiv

  9. [16]

    Kamˇ cev, S

    N. Kamˇ cev, S. Letzter, and A. Pokrovskiy,The Tur´ an density of tight cycles in three-uniform hyper- graphs, Int. Math. Res. Not. IMRN 6 (2024), 4804–4841. MR4721657

    2024

  10. [17]

    Katona, T

    G. Katona, T. Nemetz, and M. Simonovits, On a problem of Tur´ an in the theory of graphs , Mat. Lapok 15 (1964), 228–238. MR172263

    1964

  11. [18]

    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. MR2866732

    2011

  12. [20]

    X. Liu, D. Mubayi, and C. Reiher, The minimum degree of extremal hypergraphs , 2021. Manuscript

    2021

  13. [21]

    Liu and O

    X. Liu and O. Pikhurko, Finite hypergraph families with rich extremal Tur´ an constructions via mixing patterns, arXiv preprint arXiv:2212.08636 (2022)

    2022 arXiv

  14. [22]

    Mubayi, O

    D. Mubayi, O. Pikhurko, and B. Sudakov, Hypergraph Tur´ an problem: some open questions, Aim workshop problem lists, manuscript, 2011, pp. 166

    2011

  15. [23]

    Mubayi and V

    D. Mubayi and V. R¨ odl, On the Tur´ an number of triple systems , J. Combin. Theory Ser. A 100 (2002), no. 1, 136–152. MR1932073

    2002

  16. [24]

    Nagle, V

    B. Nagle, V. R¨ odl, and M. Schacht,The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms 28 (2006), no. 2, 113–179. MR2198495

    2006

  17. [25]

    Pikhurko and E

    O. Pikhurko and E. R. Vaughan, Minimum number of k-cliques in graphs with bounded independence number, Combin. Probab. Computing 22 (2013), 910–934

    2013

  18. [26]

    Pikhurko, On possible Tur´ an densities, Israel J

    O. Pikhurko, On possible Tur´ an densities, Israel J. Math. 201 (2014), no. 1, 415–454. MR3265290

    2014

  19. [27]

    Razborov, On 3-hypergraphs with forbidden 4-vertex configurations , SIAM J

    A. Razborov, On 3-hypergraphs with forbidden 4-vertex configurations , SIAM J. Discr. Math. 24 (2010), 946–963

    2010

  20. [28]

    A. A. Razborov, Flag algebras, J. Symbolic Logic 72 (2007), no. 4, 1239–1282. MR2371204 25

    2007

  21. [29]

    Reiher, V

    C. Reiher, V. R¨ odl, and M. Schacht,Embedding tetrahedra into quasirandom hypergraphs, J. Combin. Theory Ser. B 121 (2016), 229–247. MR3548293

    2016

  22. [30]

    R¨ odl and J

    V. R¨ odl and J. Skokan,Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms 25 (2004), no. 1, 1–42. MR2069663

    2004

  23. [31]

    Simonovits, A method for solving extremal problems in graph theory, stability problems , Theory of Graphs (Proc

    M. Simonovits, A method for solving extremal problems in graph theory, stability problems , Theory of Graphs (Proc. Colloq., Tihany, 1966), 1968, pp. 279–319. MR233735

    1966

  24. [32]

    Tur´ an, Eine Extremalaufgabe aus der Graphentheorie , Mat

    P. Tur´ an, Eine Extremalaufgabe aus der Graphentheorie , Mat. Fiz. Lapok 48 (1941), 436–452. MR18405 26

    1941

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.