Exact Tur\'{a}n number of the Fano plane in the $\ell_2$-norm

Jianfeng Hou; Xizhi Liu; Yixiao Zhang

REVIEW 3 major objections 3 minor 26 references

Exact Tur\'{a}n number of the Fano plane in the $\ell_2$-norm

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

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Pith's one-line read The balanced complete bipartite 3-graph is the unique extremal Fano-free 3-graph for the ℓ2-norm Turán problem, for all large n.

desk verdict Strong and likely correct, but the main theorem is conditional on a deferred unpublished lemma that carries the proof's weight. read the letter →

arxiv 2507.12354 v1 pith:G6WCCZRK submitted 2025-07-16 math.CO

Jianfeng Hou , Xizhi Liu , Yixiao Zhang This is my paper · ORCID

classification math.CO MSC 05C6505D0505C35

keywords Fanoplaneℓ2-normTuránnumberhypergraphdensityextremalstabilitybalancedcompletebipartite3-graphstarcountingmultigraphproblemvertex-extendability

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 determines the ℓ2-norm Turán problem for the Fano plane, the unique linear 3-graph on seven vertices with seven edges. For all sufficiently large n, the balanced complete bipartite 3-graph Bn is the unique Fano-free 3-graph maximizing ||H||2, the maximum equals ||Bn||2, and the corresponding Turán density is 5/16; this confirms a conjecture posed in [BCL22a]. The route is a stability theorem of the classical extremal-graph type: every Fano-free 3-graph on n vertices with minimum ℓ2-norm degree at least (5/4 − ε)n³ must be bipartite. The proof combines a new structural theorem for K4-free 5-multigraphs, a refined upper bound on two-edge stars in graphs with large high-degree independent sets, and a vertex-extendability argument showing that near-extremal graphs that become bipartite after deleting a vertex were already bipartite. A reader should care because it shows a nonlinear norm Turán problem can have exactly the same unique extremal construction as the classical edge-count problem.

What carries the argument

The argument is carried by rewriting ℓ2-norms as counts of two-edge stars. For a 3-graph H, ||H||2 = 2N(S2, H) + 3|H| and for each vertex v, d2,H(v) = 2dS2,H(v) + 3dH(v), where S2 is the 2-edge star whose center is a pair of vertices; this identity lets the minimum ℓ2-norm degree condition speak about how many S2 copies sit at each vertex. Around this identity the proof builds three tools: a stability theorem for K4-free 5-multigraphs (Theorem 2.4), which describes the possible structure of the five link graphs of a K5³ under a minimum-degree restriction; a refinement of the classical star-counting theorem of [AK78] (Proposition 2.3), which bounds two-edge stars in a graph whose edge density lies in [17/50, 7/20] and which has a large independent set of high-degree vertices; and the vertex-extendability framework of [CL24], which converts edge-stability plus vertex-extendability into the degree-stability needed for the final theorem.

What would settle it

Search for, for the value of ε claimed in Theorem 1.1, any Fano-free 3-graph on large n that is not bipartite but has δℓ2(H) ≥ (5/4 − ε)n³; finding one would disprove the main theorem. A cheaper check is to verify Proposition 2.3 numerically on graphs with edge density in [17/50, 7/20] and an independent set of the prescribed size and degrees: any counterexample to that bound would break the proof, even if the theorem itself remains true.

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Extended reading notes

Core claim

The central claim is that, for large n, the ℓ2-norm Turán number of the Fano plane F is exactly ||Bn||2, and Bn is the unique extremal construction; equivalently πℓ2(F) = 5/16. To prove this, the paper establishes the stronger statement that any F-free 3-graph on n vertices with δℓ2(H) ≥ (5/4 − ε)n³ must be bipartite. The proof first shows πℓ2(F) = 5/16: an extremal graph either avoids the complete 3-uniform hypergraph K5³ on five vertices, in which case the known value πℓ2(K5³) = 5/16 applies, or it contains a copy of K5³, in which case the five link graphs form a K4-free 5-multigraph whose structure is incompatible with a high minimum ℓ2-norm degree. Then edge-stability, vertex-extendability, and a uniqueness computation for bipartite 3-graphs upgrade the density bound to the exact unique extremal result.

Load-bearing premise

The load-bearing premise is a refined bound on two-edge stars in graphs with a large independent set of high-degree vertices, stated as Proposition 2.3 with its proof deferred to a separate paper [HLZ]; the contradiction establishing the main theorem collapses if that bound is false or unavailable, and the proof also imports, without proof, the value of the ℓ2-norm Turán density of the complete 3-graph K5³.

Editorial extensions

If this is right

For all sufficiently large n, exℓ2(n, F) = ||Bn||2, and Bn is the unique Fano-free 3-graph attaining it.
The ℓ2-norm Turán density of the Fano plane is exactly 5/16, matching the value conjectured in [BCL22a].
Every F-free 3-graph with minimum ℓ2-norm degree at least (5/4 − ε)n³ is bipartite, and near-extremal graphs are bipartite after removing o(n³) edges.
The exact value ex5(n, K4) = 2 binom(n,2) + 3 floor(n²/4) for n ≥ 632 follows from the new multigraph stability theorem.
The associated generalized Turán number N(S2, H) is uniquely maximized by Bn.

Reading between the lines

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

The same mechanism should extend to every ℓp-norm with p ≥ 1: since ||H||p is a linear combination of p-star counts via Stirling numbers, an analogue of the refined star-counting bound would likely force Bn to remain the unique extremal construction for all p; the paper proves only p = 2.
The paper's own authors note that Proposition 2.3 is proved only for edge densities x in [17/50, 7/20] but believe the bound should hold for all x > 1/4; proving that extension would simplify the case analysis in Lemma 3.6 and might reduce numerical constants such as 58/17 and 61/176.
A small-n exhaustive or SAT-based search for Fano-free 3-graphs exceeding ||Bn||2 could test how early the asymptotic extremal result begins; if no counterexample appears up to moderate n, it would support the natural guess that the exact result holds for all n ≥ 4, not merely for large n.

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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

The paper settles a real conjecture — exact ℓ2-norm Turán number of the Fano plane is 5/16 with Bn unique for large n — and the proof architecture is credible. The AES-type multigraph stability theorem (Theorem 2.4) is new and comes with a full, detailed proof in Section 5; that part looks solid and is useful on its own. Lemma 3.5 is clean, and the application of the [CL24] framework is careful. The paper is also honest about where the missing piece lives.

The problem is Proposition 2.3. It is a refinement of Ahlswede–Katona, stated without proof and deferred to an in-preparation paper [HLZ] by the same authors. It is not a sidelight: Claims 3.8–3.10 all use it to force the degree lower bounds that produce the contradiction in Lemma 3.6. The stress-test note is right that the numerical margin is thin — about 1.5×10^{-4} in units of n^2. If the proposition is false, or true but slightly weaker, the proof collapses. That makes the main theorem conditional as submitted, not wrong in logic. The imported flag-algebra result (Theorem 2.8) is published, so less of a concern, though it is still a computational black box.

My bottom line: the reader's conditional verdict is fair. The paper deserves a serious referee, but the referee should push hard for either a proof of Proposition 2.3 or a restatement of Theorem 1.1 as conditional on [HLZ]. I would bring this to a reading group, mainly to discuss the proof strategy and the multigraph theorem, and I would cite it for Theorem 2.4 and the overall approach — with a caveat on the main result. If the missing lemma appears, this becomes a strong paper; until then, treat it as a well-structured claim with a load-bearing gap.

Referee Report

3 major / 3 minor

Summary. The paper proves Theorem 1.1, an Andrásfai–Erdős–Sós-type stability theorem for Fano-plane-free 3-graphs in the ℓ2-norm: for sufficiently large n, every F-free 3-graph on n vertices with minimum ℓ2-norm degree at least (5/4 − ε)n^3 is bipartite. As a consequence, the balanced complete bipartite 3-graph Bn is the unique extremal construction for the ℓ2-norm Turán number of the Fano plane, confirming a conjecture of Balogh–Clemen–Lidický. The proof combines a refined Ahlswede–Katona bound on star counts (Proposition 2.3), a new Andrásfai–Erdős–Sós-type stability theorem for K4-free 5-multigraphs (Theorem 2.4), the published flag-algebra value πℓ2(K3_5) = 5/16 (Theorem 2.8), and the vertex-extendability framework of [CL24].

Significance. If the result holds, it confirms Conjecture 3.1 of Balogh–Clemen–Lidický and provides the first exact ℓ2-norm Turán result for a nondegenerate hypergraph family with a uniqueness statement. The paper contains substantial original components: a full proof of Theorem 2.4, an application giving the exact value of ex5(n, K4) for n ≥ 632, and a self-contained proof of uniqueness among bipartite 3-graphs (Lemma 4.6). The case analysis in Section 5 is extensive and appears structured. However, the central claim is conditional on Proposition 2.3, whose proof is explicitly deferred to an in-preparation companion paper [HLZ]; the manuscript itself states that the proof is 'rather tedious' and presented elsewhere. The published flag-algebra input [BCL22b] is less concerning but is another external dependency.

major comments (3)

[Section 2.1, Proposition 2.3] Proposition 2.3 is stated with its proof deferred to the authors' in-preparation manuscript [HLZ]. This is not a peripheral tool: the proposition is applied in Lemma 3.6, Claims 3.8, 3.9, and 3.10, and the final contradiction in Lemma 3.6 has a numerical margin of only about 1.5×10^{-4} n^2 between 5154779/2872915 and 61/34. If Proposition 2.3 is false, or is true with a slightly weaker bound on N(S2, G)/n^3, the inequalities in Claims 3.8–3.10 fail and the proof of Theorem 1.1 collapses. A complete proof of Proposition 2.3, or a reference to a publicly available version of [HLZ], is necessary before the main theorem can be considered established.
[Section 2.1, statement of Proposition 2.3] As printed, the first term of the N(S2, G) bound in Proposition 2.3 is α^3 + ((2x−α^2)(2x+α^2)^{1/2})/2, but the 'Consequently' line and the described construction Ŝ(n, αn, β3 n) correspond instead to (α^3 + (2x−α^2)(2x+α^2)^{1/2})/2 before doubling. The two readings differ by α^3, which is not negligible at the scale of the constants used in Claims 3.8–3.10. Since Proposition 2.3 is unproved and load-bearing, the statement must be corrected and proved in a consistent form; as it stands, the inequality solved in Claims 3.8–3.10 is not clearly the one supplied by the proposition.
[Lemma 3.6, Claims 3.8–3.10] In Claim 3.8, Proposition 2.3(i) is applied to Gi0 with α treated as the size parameter, but Claim 3.7 only guarantees an independent set of size at least α2 n, where α2 = 3α1/2. Since α2 ≥ α1, one can pass to a subset of size α1 n and then verify the degree condition, but this subset argument is not stated. The same issue occurs in Claim 3.9, where a subset of size α2 n (or any admissible α in [1/3, 2/5]) must be explicitly chosen. Without this clarification, the hypotheses of Proposition 2.3 are not satisfied by the sets produced in Claim 3.7 as written.

minor comments (3)

[Section 3.2, proof of Theorem 3.2] The notation δS2(G) is used before it is defined; please define it as min_{v∈V(G)} dS2,G(v) for consistency with the rest of the paper.
[Section 6, Theorem 6.3] The statement that a 'minor modification' of Lemma 4.6 yields the bound N(S2, H) ≤ N(S2, Bn) for all F-free H needs justification: Lemma 4.6 concerns ∥H∥2, and the relation ∥H∥2 = 2N(S2, H) + 3|H| involves the edge count, so the claimed transfer is not immediate. Please provide the details or mark the statement as conjectural.
[Throughout] There are several minor typographical issues, such as missing spaces in 'p = 2and r = 2' and 'Fix a vertexvn ∈ V'; these should be corrected in the final version.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular equivalence: the central derivation is not defined in terms of its target, though it leans on a deferred, load-bearing proposition.

full rationale

The proof of Theorem 1.1 is not circular in the sense of the requested patterns. The lower bound comes from the explicit balanced complete bipartite construction Bn, with δℓ2(Bn)=(5/4+o(1))n^3, and the upper bound is assembled from an external flag-algebra result (Theorem 2.8 from Balogh–Clemen–Lidický), a self-contained multigraph stability theorem (Theorem 2.4, proven in Section 5), a general vertex-extendability framework (Theorem 2.10 from Chen–Liu [CL24]), and the refined Ahlswede–Katona bound (Proposition 2.3). No fitted parameter is renamed as a prediction, and no equation is equivalent to its input by construction. The genuine gap is Proposition 2.3 in Section 2.1: it is load-bearing — Claims 3.8–3.10 in Lemma 3.6 apply it to force dmin ≥ 253n^2/730 and min{|G3|,|G4|,|G5|} ≥ 321n^2/926, with the final contradiction in Lemma 3.6 having a narrow numerical margin — yet its proof is deferred to the authors' own 'in preparation' paper [HLZ]. This is an omitted-proof/verifiability concern, not a circular reduction: the proposition is a parameter-free inequality about graphs and does not mention the Fano plane, and the main theorem still has substantial independent content (the multigraph stability theorem and vertex-extendability proof). The self-citation to [CL24] is likewise load-bearing but supplies a general framework whose stated assumptions do not include the target result. Overall, the paper's derivation does not reduce to its own inputs; the score reflects the unverified self-referential black box, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof rests on several external results: the classical AES theorem, the Ahlswede-Katona theorem, the Bellmann-Reiher bound for 5-multigraphs, a flag-algebra result on K3_5 from BCL22b, and the stability framework of CL24/CIL+24. The paper's own Proposition 2.3 is stated without proof (deferred to [HLZ]) and acts as an unproven axiom in this manuscript. No free parameters are fitted to data; all constants are existential and fixed by the proof.

assumptions (6)

standard math Theorem 2.1 (Andrásfai-Erdős-Sós): every K3-free graph on n vertices with minimum degree > 2n/5 is bipartite.
Used in Claim 5.13 to show the graph H of pairs with multiplicity ≥ 3 is bipartite.
standard math Theorem 2.2 (Ahlswede-Katona bound on stars), as stated.
Used in Lemma 3.3 to lower-bound degree from ℓ2-norm degree.
ad hoc to paper Proposition 2.3 (refined Ahlswede-Katona bound under independent-set degree conditions).
Stated without proof; deferred to [HLZ] (in preparation). Used in Claims 3.8-3.10 and in Section 4.
domain assumption Theorem 2.8: πℓ2(K3_5)=5/16 and associated stability, from [BCL22b].
Imported as a black box; relies on flag algebra computations.
domain assumption Theorem 2.10 (ℓp-degree-stability from ℓp-edge-stability plus vertex-extendability), from [CL24].
Framework used to derive degree-stability from edge-stability and vertex-extendability.
standard math Bellmann-Reiher bound (5): ex5(n, K4) ≤ (7n^2 - n)/4.
Published result used in Lemma 3.4 and Theorem 6.1.

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Pith. "Pith review of Exact Tur\'{a}n number of the Fano plane in the $\ell_2$-norm." pith.science (2026). https://pith.science/paper/G6WCCZRK

@misc{pith2026250712354,
  author       = {Pith},
  title        = {Pith review of: Exact Tur\'an number of the Fano plane in the $\ell_2$-norm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G6WCCZRK}},
  note         = {Machine review of arXiv:2507.12354}
}

abstract

A classical object in hypergraph Tur\'{a}n theory is the Fano plane $\mathbb{F}$, the unique linear $3$-graph on seven vertices with seven edges. The Tur\'{a}n density and exact Tur\'{a}n number of $\mathbb{F}$, first proposed as a problem by S\'{o}s \cite{Sos76} in the 1970s, were determined through a sequence of works by De Caen-F\"{u}redi \cite{DCF00}, F\"{u}redi-Simonovits \cite{FS05}, Keevash-Sudakov \cite{KS05}, and Bellmann-Reiher \cite{BR19}. Addressing a conjecture of Balogh-Clemen-Lidick\'{y} \cite[Conjecture 3.1]{BCL22a}, we establish an Andr\'{a}sfai-Erd\H{o}s-S\'{o}s-type stability theorem for $\mathbb{F}$ in the $\ell_2$-norm: there exists a positive constant $\varepsilon$ such that for large $n$, every $\mathbb{F}$-free $3$-graph on $n$ vertices with minimum $\ell_2$-norm degree at least $(5/4 - \varepsilon)n^3$ must be bipartite. As a consequence, for large $n$, the balanced complete bipartite $3$-graph is the unique extremal construction for the $\ell_{2}$-norm Tur\'{a}n problem of $\mathbb{F}$, thereby confirming the conjecture of Balogh-Clemen-Lidick\'{y}. Our proof includes a refinement of a classical result by Ahlswede-Katona \cite{AK78} on counting stars, and the establishment of an Andr\'{a}sfai-Erd\H{o}s-S\'{o}s-type theorem for a multigraph Tur\'{a}n problem studied by Bellmann-Reiher \cite{BR19}, both of which are of independent interest.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[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

work page 1974

[2] [2]

Ahlswede and G

R. Ahlswede and G. O. H. Katona. Graphs with maximal number of adjacent pairs of edges. Acta Math. Acad. Sci. Hungar. , 32(1-2):97--120, 1978

work page 1978

[3] [3]

Hypergraph T ur\' a n problems in _2 -norm

J\' o zsef Balogh, Felix Christian Clemen, and Bernard Lidick\' y . Hypergraph T ur\' a n problems in _2 -norm. In Surveys in combinatorics 2022 , volume 481 of London Math. Soc. Lecture Note Ser. , pages 21--63. Cambridge Univ. Press, Cambridge, 2022

work page 2022

[4] [4]

Solving T ur\'an's tetrahedron problem for the _2 -norm

J\'ozsef Balogh, Felix Christian Clemen, and Bernard Lidick\'y. Solving T ur\'an's tetrahedron problem for the _2 -norm. J. Lond. Math. Soc. (2) , 106(1):60--84, 2022

work page 2022

[5] [5]

Some exact and asymptotic results for hypergraph T ur\' a n problems in _2 -norm

George Brooks and William Linz. Some exact and asymptotic results for hypergraph T ur\' a n problems in _2 -norm. arXiv preprint arXiv:2310.09379 , 2023

arXiv 2023

[6] [6]

Degree powers in graphs: the E rd o s- S tone theorem

B\'ela Bollob\'as and Vladimir Nikiforov. Degree powers in graphs: the E rd o s- S tone theorem. Combin. Probab. Comput. , 21(1-2):89--105, 2012

work page 2012

[7] [7]

Tur\'an's theorem for the F ano plane

Louis Bellmann and Christian Reiher. Tur\'an's theorem for the F ano plane. Combinatorica , 39(5):961--982, 2019

work page 2019

[8] [8]

Generalized A ndr \' a sfai-- E rd o s-- S \'o s theorems for odd cycles

Zian Chen, Jianfeng Hou, Caiyun Hu, and Xizhi Liu. Generalized A ndr \' a sfai-- E rd o s-- S \'o s theorems for odd cycles. arXiv preprint arXiv:2409.11950 , 2024

arXiv 2024