Spectral generalized Tur\'{a}n problems

Xizhi Liu

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Spectral generalized Tur\'{a}n problems

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Pith's one-line read This paper introduces the spectral generalized Turán problem—maximizing the $(\alpha,Q)$-spectral radius among $F$-free hypergraphs—and proves a general stability-to-spectrum theorem whose first application is the spectral Erdős Pentagon…

desk verdict New framework for spectral generalized Turán problems with a transfer theorem, but the proof has a few real gaps, including a reversed inequality in Lemma 3.3. read the letter →

arxiv 2507.21689 v1 pith:HQ244CCO submitted 2025-07-29 math.CO

Xizhi Liu This is my paper · ORCID

classification math.CO MSC 05C3505C6505C50

keywords spectralTuránproblemgeneralized(αQ)-spectralradiusErdősPentagonTheoremdegreestabilityentropicdensityhypergraphs

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 sets up a new combined extremal problem: instead of counting copies of a fixed $r$-graph $Q$ in an $F$-free hypergraph, or taking the classical spectral radius, it maximizes the $(\alpha,Q)$-spectral radius—the maximum of the Lagrangian $Q$-polynomial over vectors of unit $\ell^\alpha$ norm—among $F$-free $r$-graphs on $n$ vertices. The main theorem (Theorem 1.5) shows that if the forbidden family is 'smooth' and 'degree stable' with respect to a hereditary family $\mathcal H$ of $F$-free $r$-graphs, and $\mathcal H$ is 'balanced in spectral', then for all large $n$ every $F$-free $r$-graph has $(\alpha,Q)$-spectral radius at most the maximum inside $\mathcal H$, with equality only there. This determines the spectral generalized Turán number and extends the classical spectral Turán theorem of [KLM14] from edges to arbitrary $Q$. A short application is the spectral Erdős Pentagon Theorem: large triangle-free graphs maximize their $C_5$-spectral radius only on $C_5$-colorable graphs. The paper also proves that the $(\alpha,Q)$-spectral radius equals an entropic density defined by random embeddings of $Q$, extending a recent identity of [CY24].

What carries the argument

The central object is the $(\alpha,Q)$-spectral radius $\lambda_{\alpha,Q}(H)$: the maximum, over vectors $x$ with $\sum_i|x_i|^\alpha=1$, of the Lagrangian $Q$-polynomial $P_{Q,H}(x)=\sum_{\phi\in\mathrm{Inj}(Q,H)}\prod_{i\in\phi(V(Q))}x_i$. The argument is carried by a vertex-removal iteration. The Lagrange multiplier rule (Lemma 3.1) equates partial derivatives of $P_{Q,H}$ at an optimal vector with $q\lambda_{\alpha,Q}(H)x_i^{\alpha-1}$; from this, Lemma 3.3 shows that a graph with large spectral radius but small minimum $Q$-degree must have an optimal vector with a very small coordinate, and Lemma 3.5 shows that deleting that vertex barely reduces the spectral radius. Iterating, the spectral radius is driven down until it contradicts the asymptotic value forced by the spectral balance condition. In the pentagon application, the proof translates $C_5$-copies in a $C_5$-colorable graph into edges of a 5-partite 5-graph and bounds its spectral radius with the $m$-partite bound of [KNY15] together with an estimate for the balanced complete 5-partite 5-graph.

What would settle it

Compute, for $\alpha=2$ and increasing $n$, the maximum $(\alpha,C_5)$-spectral radius among triangle-free graphs on $n$ vertices and compare it with the maximum over $C_5$-colorable graphs; an infinite sequence of triangle-free graphs whose value strictly exceeds every $C_5$-colorable graph on the same $n$ would refute the spectral Erdős Pentagon Theorem. A cheaper check on the machinery is to test Claim 4.4 numerically: the difference $|\lambda_{\alpha,C_5}(n,\mathcal C_5)-|\mathrm{Aut}(C_5)||T^5_{5,n}|/n^{5/\alpha}|$ must grow at most like $n^{4-5/\alpha}$; faster growth would invalidate the balance verification.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.5. Let $\alpha>1$ and $\varepsilon>0$ be real numbers, $Q$ an $r$-graph on $q$ vertices, $F$ a family of $r$-graphs with positive embedding density $\hat\pi(Q,F)$, and $\mathcal H$ a hereditary family of $F$-free $r$-graphs. Suppose $F$ is $(\delta,M,Q)$-smooth, $F$ is $(\varepsilon,M,Q)$-degree stable with respect to $\mathcal H$, and $\mathcal H$ is $(\alpha,\delta,M,Q,F)$-balanced in spectral. Then every $F$-free $r$-graph $H$ on $n\ge N$ vertices satisfies $\lambda_{\alpha,Q}(H)\le\lambda_{\alpha,Q}(n,\mathcal H)$, with equality only if $H\in\mathcal H$; hence $\mathrm{spex}_\alpha(n,Q,F)=\lambda_{\alpha,Q}(n,\mathcal H)$ for all large $n$. Smoothness controls the growth of the injection count $\mathrm{inj}(n,Q,F)$, degree stability says the family of near-extremal graphs in $Q$-degree is inside $\mathcal H$, and spectral balance pins the maximum $(\alpha,Q)$-spectral radius inside $\mathcal H$ to the asymptotic embedding count $\mathrm{inj}(n,Q,F)/n^{q/\alpha}$ up to an error of order $n^{q-q/\alpha-1}$. The paper verifies these hypotheses for $Q=C_5$, $F=\{K_3\}$, and $\mathcal H$ the $C_5$-colorable graphs, using the combinatorial Erdős Pentagon Theorem and an $m$-partite spectral bound, and obtains the spectral Erdős Pentagon Theorem. It also proves the identity $\eta_{\alpha,Q}(H)=\lambda_{\alpha,Q}(H)$ for the newly defined $(\alpha,Q)$-entropic density.

Load-bearing premise

The spectral balance condition (Definition 1.4(iii))—that the maximum $(\alpha,Q)$-spectral radius inside the hereditary extremal family $\mathcal H$ matches $\mathrm{inj}(n,Q,F)/n^{q/\alpha}$ up to an error of order $n^{q-q/\alpha-1}$—is the load-bearing premise; if it fails, Theorem 1.5 does not apply, and in the pentagon application it is established by combining the full combinatorial Erdős Pentagon Theorem with the $m$-partite spectral bound, so it is a substantial hypothesis rather than a formality.

Editorial extensions

If this is right

For any family $F$ and pattern $Q$ satisfying smoothness, $Q$-degree stability, and spectral balance, the spectral generalized Turán number $\mathrm{spex}_\alpha(n,Q,F)$ is simply the maximum $(\alpha,Q)$-spectral radius inside the stable family $\mathcal H$, for all large $n$.
Under the spectral Erdős Pentagon Theorem, for $\alpha>1$ and large $n$, every triangle-free graph $G$ has $\lambda_{\alpha,C_5}(G)\le\max_{H\in\mathcal C_5,\,v(H)=n}\lambda_{\alpha,C_5}(H)$, with equality only for $C_5$-colorable $G$.
Because $\eta_{\alpha,Q}(H)=\lambda_{\alpha,Q}(H)$, spectral extremal problems for a fixed pattern $Q$ can be reformulated as entropy maximization over random embeddings of $Q$ in $F$-free hypergraphs.
The framework turns any degree-stability theorem for a generalized Turán problem into a spectral statement, provided the spectral balance condition is verified, thereby extending the reach of the classical reduction of [KLM14].

Reading between the lines

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

The spectral balance condition is the genuine bottleneck: it ties the spectral maximum inside $\mathcal H$ to a purely combinatorial embedding count, so for most natural problems verifying it may be as hard as the spectral problem itself; the pentagon case succeeds only because the full combinatorial Erdős Pentagon Theorem supplies the extremal family. This is an editorial reading of where the dif
The entropy identity suggests a testable strategy for open generalized Turán problems: maximize $\eta_{\alpha,Q}$ over $F$-free hypergraphs through entropy inequalities, then read off the spectral extremal value; this could bypass flag-algebra computations in cases where balanced families are known.
Since the combinatorial pentagon theorem holds for every $n$, a natural extension is to ask whether the spectral pentagon theorem also holds for all $n$ and for $\alpha=1$, and whether similar transfers work for other odd cycles using the degree-stability results of [CHHL24].

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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 opens the spectral generalized Turán direction and proves a transfer theorem that converts smoothness, degree stability, and spectral balance into a spectral extremal result, generalizing Keevash–Lenz–Mubayi. That is genuinely new and useful. The pentagon application is clean and the entropy equivalence is a nice extension of Chao–Hans. The setup is careful and the writing is mostly clear.

The proofs have a few real soft spots. The most serious is in Lemma 3.3: the proof claims (1−ε)^((α−1)/α) ≤ (1−ε), which is reversed for α>1. This means the contradiction is not established as written. The gap looks repairable by choosing δ smaller than πhat ε/((q−1)α), specifically satisfying 1−αδ > (1−ε)^((α−1)/α), but the manuscript does not do this. The stress-test note is correct.

There is also an issue near the end of Theorem 1.5. The text applies Fact 2.5 with δ′=1/2 and then asserts μ_n ≥ πhat n^{q−q/α}. Fact 2.5 only gives μ_n ≥ (πhat − δ′)n^{...}, so you cannot just drop the factor. This is likely a minor fix—use a small δ′ and keep the (1−δ′) factor—but as written it is wrong.

The entropy proof in Proposition 5.1 has the stationarity equation on the support only; the text writes it as if it holds for all classes. The subsequent inequality still goes through if you define β over the support and note that β ≤ P_{Q,H}(x), so this is a minor presentational flaw, not a logical one.

No circularity, no data issues. The main ideas are sound and the theorems are plausible. The gaps are in the execution, not the architecture. This is the kind of paper that belongs in the refereeing pipeline—it will need a revision, but it deserves a serious referee.

The paper is for extremal combinatorics readers who work on spectral Turán problems or generalized Turán problems. I would not cite it in the next year unless I were working directly on spectral extremal problems, but I would bring it to a reading group.

Referee Report

2 major / 4 minor

Summary. The paper introduces the spectral generalized Turán problem: for an r-graph Q and a family F, it studies the maximum of the (α,Q)-spectral radius λ_{α,Q}(H) over F-free r-graphs H on n vertices. The main result (Theorem 1.5) asserts that under three hypotheses on F and a hereditary family H—smoothness of inj(n,Q,F), Q-degree stability of F with respect to H, and spectral balance of H—every F-free r-graph H on n vertices satisfies λ_{α,Q}(H) ≤ λ_{α,Q}(n,H), with equality only when H∈H; consequently spex_α(n,Q,F)=λ_{α,Q}(n,H). Theorem 1.2 derives the spectral Erdős Pentagon Theorem from this framework. Section 5 defines the (α,Q)-entropic density and proves η_{α,Q}(H)=λ_{α,Q}(H), extending a result of Chao and Hans.

Significance. The framework is natural and the main theorem, if correct, provides a useful transfer principle: it reduces spectral generalized Turán problems to three extremal/structural quantities, in the spirit of Keevash–Lenz–Mubayi. The pentagon application is a genuine new result and combines the Erdős Pentagon Theorem with the Kang–Nikiforov–Yuan spectral bound in a non-obvious way. The entropy identity is elegant and appears essentially correct. The paper is not accompanied by code or machine-checked proofs, but the argument is classical analytic/combinatorial in style. The proof gaps identified below are substantial, though each appears local and repairable.

major comments (2)

[Section 3.1, Lemma 3.3] The display after Claim 3.4 uses the inequality (1−ε)^((α−1)/α) ≤ (1−ε). For α>1 and ε∈(0,1), this inequality is reversed: since the exponent (α−1)/α is smaller than 1, (1−ε)^((α−1)/α) is larger than (1−ε). Consequently the claimed upper bound in the contradiction is not established. The intended contradiction requires (1−αδ) > (1−ε)^((α−1)/α) with δ = πhat ε/((q−1)α), which in the small-ε limit becomes πhat/(q−1) < (α−1)/α. No such condition appears in Theorem 1.5, and it fails in the pentagon application for α close to 1. Since Lemma 3.3 produces the first small coordinate for the iterative deletion, the proof of Theorem 1.5 is incomplete as written. The lemma appears repairable by choosing a different tolerance, but the repair must be supplied.
[Section 3.2, proof of Theorem 1.5, final displayed lower bound] Fact 2.5 is invoked with δ′ replaced by 1/2 in order to assert μ_n ≥ πhat n^{q−q/α}. Fact 2.5 gives only μ_n ≥ (πhat−1/2)n^{q−q/α}, which is useless and even negative when πhat is small; in the pentagon application πhat≈10/5^5≈0.0032. The correct choice would be δ′=πhat/2 or another positive constant smaller than πhat, after which the displayed product lower bound still yields a contradiction provided condition (19) is adjusted accordingly. As written, this is a second load-bearing gap in the final step of the iteration.

minor comments (4)

[Fact 1.3] The first displayed inequality claims λ_{α,Q}(H) ≥ |H| n^{−q/α} for arbitrary Q; this is false, since |H| is the number of edges. The correct inequality is λ_{α,Q}(H) ≥ inj(Q,H)n^{−q/α}. For Q=K_r^r the two statements coincide up to the factor r!, which is the case actually used in Lemma 4.1, but the stated general fact should be corrected.
[Theorem 1.2 and Section 1.1] The notation λ_α(G) is used in the statement of Theorem 1.2 and in the surrounding discussion, but the quantity being bounded is the (α,C5)-spectral radius; it should be written λ_{α,C5}(G), or the abbreviation should be defined explicitly.
[Section 5, Claim 5.2] The claim says "Let (X_1,...,X_n) be the random embedding of Q in H", but the tuple should have length q, not n. In the same proof, terms of the form y_j log(x_j^α/y_j) with y_j=0 and x_j=0 require a convention; the paper only sets 0·log 0=0, which does not directly cover the ratio 0/0.
[Fact 2.5] In the proof of Fact 2.5, the transition δ/n + δ′/2 ≤ δ′ is not immediate as written; it requires choosing n large enough so that δ/n ≤ δ′/2. This is a minor omission, but it should be stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.5 is a transfer result whose hypotheses do not encode its conclusion, and the cited stability/spectral facts are independent.

full rationale

The derivation chain is not circular. Theorem 1.5 assumes (i) smoothness, (ii) degree stability with respect to H, and (iii) spectral balance of H. Assumption (iii) only says that the maximum (α,Q)-spectral radius inside H is asymptotically inj(n,Q,F)n^{-q/α}; it does not assert that every F-free graph lies below λα,Q(n,H). The proof supplies that missing upper bound: assuming an F-free graph with λα,Q(H) ≥ µn, degree stability forces small Q-degree, Lemma 3.3 produces a small coordinate, Lemma 3.5 deletes that vertex while keeping spectral radius above µ_{m}, and iteration yields an impossible graph on N vertices. Fact 2.5 is used only to estimate µn from (iii), not to assume the inequality being proved. The pentagon application verifies the three hypotheses from external results: Theorem 1.1 gives smoothness and the injective-count value, Lemma 4.1/KNY15 give the m-partite spectral bound, and [CL24, Theorem 2.3] supplies degree stability. [CL24] is a prior independent stability theorem by one of the authors; it is a real external proof and is not an assumption of the spectral conclusion, so it does not make the argument circular. The entropy section proves ηα,Q = λα,Q directly in both directions (Claim 5.2 and the Lagrange-multiplier argument), rather than merely renaming the spectral radius. The only substantive concern apparent in the written proof (the direction of the (1−ε)^{(α−1)/α} estimate in Lemma 3.3) is a correctness issue in the deletion step, not a circularity: no displayed equation in the paper is equivalent by construction to a fitted parameter or to the theorem's conclusion. Hence no circular step is present.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

No empirical constants are fitted. All parameters such as α, δ, ε, M, N are variables in universal statements and their existence is proved. The paper introduces formal definitions, spectral generalized Turán numbers, the (α,Q)-spectral radius, and entropic density, but no empirical entities such as new particles, forces, or dimensions.

assumptions (7)

standard math Lagrange multiplier method applies to the constrained maximum of P_{Q,H} over the ℓ_α unit sphere.
Used in Lemma 3.1 and in Proposition 5.1 to characterize stationary points; standard in spectral hypergraph theory.
standard math Fact 2.4: for smooth families, inj(n,Q,F) = πhat(Q,F)n^q + o(n^q).
Follows from the definition of πhat as a limit; used throughout the proof to replace finite-n counts by their asymptotic form.
domain assumption Kang-Nikiforov-Yuan bound (Theorem 4.2): the complete balanced m-partite q-graph maximizes the α-spectral radius among m-partite q-graphs, and the Hölder-type inequality (26) holds.
External theorem from [KNY15], used in the proof of Lemma 4.1 and the pentagon application.
domain assumption Erdős Pentagon Theorem (Theorem 1.1): ex(n,C5,K3) is achieved by C5-colorable graphs.
External theorem from [Grz12, HHK+13, LP18], used to identify inj(n,C5,K3) with |Aut(C5)||T^5_{5,n}|.
domain assumption Degree stability of K3-free graphs with respect to C5-colorable graphs ([CL24, Theorem 2.3]).
External prior result, from the same author group, supplying the stability hypothesis (ii) in the pentagon application.
standard math Basic inequalities: Bernoulli, (1-x)^{-β} ≥ 1+βx, and 1-x ≥ e^{-x-x^2}.
Proved in Appendix A; used in Lemmas 3.2, 3.5, and the final product estimate in Theorem 1.5.
standard math Convexity of t log t and the log-sum inequality.
Used in Proposition 5.1 to bound entropy differences and to show η_{α,Q}(H) ≥ λ_{α,Q}(H).

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Pith. "Pith review of Spectral generalized Tur\'{a}n problems." pith.science (2026). https://pith.science/paper/HQ244CCO

@misc{pith2026250721689,
  author       = {Pith},
  title        = {Pith review of: Spectral generalized Tur\'an problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HQ244CCO}},
  note         = {Machine review of arXiv:2507.21689}
}

read the original abstract

Combining two well-studied variants of the classical Tur\'{a}n problem, the generalized Tur\'{a}n problem and the spectral Tur\'{a}n problem, we introduce the spectral generalized Tur\'{a}n problem and establish a general theorem that extends the result of Keevash--Lenz--Mubayi~\cite{KLM14} on the spectral Tur\'{a}n problem in this broader setting. As a quick application, we obtain the spectral Erd\H{o}s Pentagon Theorem. We also introduce the notion of entropic density for generalized Tur\'{a}n problems, and show that it coincides with the generalized spectral radius, extending a recent result of Chao--Hans on entropic Tur\'{a}n density.

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Pith tools

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

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