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Spectral Tur\'an-type problems for the $\alpha$-spectral radius of hypergraphs with degree stability

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proves that degree-stable forbidden families of hypergraphs have spectral Turán problems reducible to pattern-colorable families, and uses this to identify the unique extremal hypergraph for r-expansions of color-critical graphs.

desk verdict The reduction idea is good and the regularity theorem is a real step forward, but the edge-extremal equality claims don't follow from the α→∞ argument, and the main induction has a base-case gap. read the letter →

arxiv 2509.24354 v2 pith:FRFBLHQL submitted 2025-09-29 math.CO

Jian Zheng , Honghai Li , Li Su This is my paper · ORCID
classification math.CO MSC 05C6505C5005C35
keywords spectralTurán-typeproblemsα-spectralradiushypergraphdegreestabilityr-patternP-colorabler-expansioncolor-criticalgraph
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 develops a general reduction for spectral Turán-type problems in r-uniform hypergraphs. Its central theorem states that if a forbidden family F is degree-stable with respect to the family Col(P) of P-colorable r-graphs for some r-pattern P, then for all sufficiently large n every n-vertex F-free r-graph G satisfies λ^(α)(G) ≤ λ^(α)(Col(P), n), and equality forces G to be P-colorable. This reduces the spectral extremal problem for F to the spectral extremal problem inside Col(P). The reduction is powered by a new asymptotic-regularity theorem showing that spectral extremal hypergraphs in Col(P) have minimum eigenvector component and minimum degree close to uniform bounds. As the headline application, the authors prove that for the r-expansion F^(r) of an (l+1)-color critical graph, the balanced complete l-partite r-graph T_l^r(n) is the unique maximizer of the α-spectral radius among F^(r)-free r-graphs, and the same machinery yields the exact edge Turán number e(T_l^r(n)).

What carries the argument

The argument rests on three pieces. (i) Theorem 3.10, an asymptotic-regularity theorem for spectral extremal hypergraphs in Col(P): for α>1, the minimum component of a principal eigenvector satisfies (x_min)^α ≥ (1/n)(1−O(1/n)), and the minimum degree satisfies δ(G_n) ≥ π(Col(P))(1−O(1/n)) binom(n,r−1). (ii) The paper's degree-stability notion (Definition 4.1): a family F is degree-stable with respect to Col(P) if every F-free r-graph with minimum degree within ε of the Turán density is P-colorable. (iii) A one-vertex extension lemma (Claim 1 in Theorem 3.10) that propagates the eigenvector lower bound from n to n+1, enabling induction that the spectral extremizer of the F-free family lies i

What would settle it

Compute the α-spectral radius for α=2 of the 3-uniform expansion of a 5-cycle, C_5^(3), on n=9 vertices and compare it with λ^(2)(T_2^3(9)); if any C_5^(3)-free 3-graph on 9 vertices has larger value, Theorem 4.8 fails. More systematically, a computational search over all F^(r)-free r-graphs on small n for a given color-critical F would reveal whether T_l^r(n) is the unique maximizer; the first counterexample would localize the failure of the imported degree-stability table.

Watch

Extended reading notes

Core claim

The core discovery is that degree stability functions as a bridge from spectral Turán problems to pattern-coloring problems. Theorem 4.2 states: if F is degree-stable with respect to Col(P), then for sufficiently large n, every n-vertex F-free r-graph G obeys λ^(α)(G) ≤ λ^(α)(Col(P), n), with equality only when G is P-colorable. The proof shows the spectral extremal F-free hypergraph inherits a minimum-degree lower bound sufficient to trigger the degree-stability hypothesis, thereby forcing membership in Col(P). The advertised application, Theorem 4.8, states that for an (l+1)-color critical graph F and α≥1, the maximum λ^(α) among n-vertex F^(r)-free r-graphs is exactly λ^(α)(T_l^r(n)); for

Load-bearing premise

The proof imports, without proof, the fact that the r-expansion of any (l+1)-color critical graph is degree-stable with respect to Col(K_l^r), as tabulated in an external preprint; if that degree-stability assertion is false for some expansion, the conclusion that the extremal hypergraph is P-colorable — and hence that T_l^r(n) is extremal — no longer follows.

Editorial extensions

If this is right

  • For any degree-stable F, the spectral Turán problem for F collapses to the spectral extremal problem in Col(P): λ^(α)(M on(F), n) = λ^(α)(Col(P), n) for all large n.
  • For F the r-expansion of an (l+1)-color critical graph, the unique α-spectral extremizer (α>1) is T_l^r(n), and Corollary 4.5 gives λ^(α)(M on(F), n) = π(Col(P)) n^{r−r/α} − O(1) n^{r−r/α−1}.
  • The same reduction yields edge Turán results: every F-free n-vertex r-graph has at most ex(Col(P), n) edges, and equality forces P-colorability; for r-expansions this gives the exact Turán number e(T_l^r(n)).
  • Theorem 3.12 shows spectral extremal k-chromatic r-graphs are asymptotically balanced, partially addressing a conjecture on k-chromatic r-graphs.
  • The method is a template: any degree-stable family automatically inherits both spectral and edge extremal theorems with the same extremal structure.

Reading between the lines

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

  • The reduction likely extends to other families for which degree stability is known or can be proved, such as cancellative 3-graphs or Fano-plane-free hypergraphs, yielding new spectral Turán results.
  • The O(1/n) regularity bounds in Theorem 3.10 suggest the extremal spectral radius has a well-defined second-order term; determining it might give sharper asymptotic formulas for λ^(α)(M on(F), n).
  • If the authors' Problem 5.6 has an affirmative answer, then the spectral and edge extremal sets for Col(P) coincide, which combined with Theorem 4.2 would characterize the extremal F-free hypergraphs completely.
  • A natural test of the color-critical hypothesis: check whether the uniqueness of T_l^r(n) persists when F is only (l+1)-chromatic but not color-critical; a counterexample would show the critical-coloring condition is essential.
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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

3 major / 4 minor

Summary. The paper develops a reduction framework for spectral Turán-type problems for the α-spectral radius of r-uniform hypergraphs. It defines r-patterns P and the family Col(P) of P-colorable hypergraphs, proves asymptotic regularity of spectral extremal hypergraphs in Col(P) (Theorem 3.10), and then proves that if a forbidden family F is degree-stable with respect to Col(P), every F-free r-graph G satisfies λ^(α)(G) ≤ λ^(α)(Col(P), n) (Theorem 4.2). The paper applies this to r-expansions of color-critical graphs (Theorem 4.8), using a degree-stability result imported from [12], and derives edge-Turán results by taking α → ∞ in Section 5. A further theorem (Theorem 3.12) shows that spectral extremal k-chromatic r-graphs are asymptotically balanced, giving partial information toward a conjecture of Kang–Nikiforov–Yuan.

Significance. If valid, the spectral reduction is genuinely useful: it converts spectral Turán problems for degree-stable forbidden families into problems about pattern-colorable families, and Theorem 3.10 provides quantitative regularity information (eigenvector lower bounds and minimum-degree lower bounds) for a broad class of hypergraphs. The paper also includes a self-contained appendix for some foundational lemmas and clearly identifies Col(P) as hereditary and multiplicative. However, the advertised equality characterizations — both in the spectral theorem and in the edge-Turán application — are not fully justified by the proofs as written, and the headline concrete application depends on an unproved external table from an arXiv preprint. These issues affect load-bearing claims, though they appear to be local and potentially repairable.

major comments (3)
  1. [Theorem 5.1, Section 5] The limiting argument in the proof of Theorem 5.1 proves only the upper bound e(G) ≤ ex(Col(P), n). From r!e(G)=lim_{α→∞} λ^(α)(G) = lim_{α→∞} λ^(α)(Col(P), n) one cannot infer λ^(α)(G) = λ^(α)(Col(P), n) for any finite α, so the 'Moreover' clause cannot be obtained by invoking Theorem 4.2's equality case. Consequently the equality characterizations in Corollaries 5.3 and 5.4 are unsupported. A separate argument — for example, showing that an edge-extremal F-free graph has high minimum degree after deleting o(n) vertices, and then applying degree stability — is needed but absent.
  2. [Theorem 4.2] The induction in the proof shows only that the particular chosen maximizer H_n is P-colorable for all n ≥ n1. It does not show that an arbitrary F-free G satisfying λ^(α)(G)=λ^(α)(Mon(F), n) is P-colorable. Thus the assertion 'equality holds only if G is P-colorable' is not established. This equality clause is used in Corollaries 4.7 and 4.8 via Lemma 4.6, and also in Corollary 4.4. To complete the proof one must prove that every spectral extremizer in Mon(F), not merely one member of a chosen sequence, has the eigenvector/min-degree properties; this is missing.
  3. [Theorem 4.8 / §4] The proof of Theorem 4.8 invokes 'As established in Table 1 of [12]' for the degree-stability of the r-expansion of an (l+1)-color critical graph with respect to Col(K_l^r). This is a load-bearing external input from an arXiv preprint and is not reproduced or proved in the manuscript. If that table is unavailable or incorrect, the applications Theorems 4.8 and Corollary 5.4 do not follow from the present work. The authors should state the exact stability theorem used and either prove it or make the citation self-contained.
minor comments (4)
  1. [Lemma 3.6 / Theorem 3.10(2)] The proof of Theorem 3.10(2) cites [4, Theorem 4.6] for the statement that any hereditary and multiplicative family is clonal. The theorem is not quoted. Please state the precise result from [4] or give a proof in the appendix.
  2. [Corollary 4.4] 'Principle eigenvector' should be 'principal eigenvector'.
  3. [References] There are several reference typos: [30] lists pages '183-179' (should be '179-183'); [8] lists year '1996' (should be '1966'); [37] gives 'N. Sergey, Y. Liana' — the authors are S. Norin and L. Yepremyan.
  4. [Theorem 3.10(1)] The construction H_n∘k is introduced only inline and would benefit from a formal definition and a brief explanation of why it is edge-maximal and P-colorable.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity; the main spectral reduction is self-contained. The Section 5 equality characterization is an omitted proof rather than a circular step.

  1. other [Section 5, Theorem 5.1 proof; also Corollaries 5.3–5.4]
    "By Theorem 4.2, for any α>1 and n sufficiently large, we have λ^(α)(G)≤λ^(α)(Col(P), n) = max_{H∈Col(P)_n} λ^(α)(H), and thus, r!e(G) = lim_{α→∞} λ^(α)(G)≤ lim_{α→∞} max_{H∈Col(P)_n} λ^(α)(H) = max_{H∈Col(P)_n} lim_{α→∞} λ^(α)(H) = max_{H∈Col(P)_n} r!e(H). This implies e(G)≤ex(Col(P), n), completing the proof."

    The promised 'Moreover, if equality holds then G is P-colorable' clause is not derived. The displayed limiting argument proves only the upper bound e(G)≤ex(Col(P),n). If e(G)=ex(Col(P),n), the two α→∞ limits are equal, but equality of limits does not imply λ^(α)(G)=λ^(α)(Col(P),n) for any finite α, which is exactly what Theorem 4.2's equality case would require. Thus the equality characterization in Theorem 5.1, and the same 'letting α→∞' inference in Corollaries 5.3–5.4, are unsupported by the paper's own equations. This is an omitted-proof/non-sequitur issue, not a fit or self-citation chain, so it is not circularity in the sense of this review.

full rationale

The central spectral reduction, Theorem 4.2, is not circular. Its proof uses the degree-stability hypothesis as an external input and separately proves that spectral extremal graphs have high minimum degree via Lemmas 3.2/3.3 and Theorem 3.10. Those ingredients rely on independent prior results (Nikiforov [28,29], Kang–Nikiforov–Yuan [18], Cooper–Desai–Sahay [4]), not on the conclusion being proved. No parameter is fitted to the target value, and no object is defined in terms of the quantity it is supposed to predict. The application Theorem 4.8 depends on the external degree-stability table of [12], which is a citation to other authors' work, not a self-citation; even if that input were unverified, it would be a correctness risk rather than circularity. The only self-citation, [47], appears in the introduction as context and is not used in any proof. The flagged Section 5 issue is a genuine logical gap: the equality case of the edge-Turán characterization is asserted after a limiting argument that cannot deliver it. However, this is a non-sequitur, not a circular derivation, so the circularity score remains low.

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

The paper introduces no fitted constants or new postulated objects. Its free parameters are structural constants (epsilon, M, M0) chosen for the proofs rather than fit to data. The axioms are standard analytic inequalities plus cited structural theorems from [4], [12], [18], and [28,29], which the paper does not re-derive.

assumptions (6)
  • standard math Nikiforov's analytic Perron-Frobenius theory and flatness lemmas for the alpha-spectral radius (Lemmas 2.1-2.3, 2.7-2.9 from [28,29]).
    Used throughout: eigenequation (2.2), limits (2.4), equality of spectral and edge densities for alpha>1, and the bound lambda <= pi n^{r-r/alpha}.
  • domain assumption Col(P) is hereditary and multiplicative (Fact A in Appendix).
    This is proved in the appendix, but it is the structural assumption that makes flatness and the clonal-family machinery applicable.
  • domain assumption Cooper-Desai-Sahay clonal-family theorem: every hereditary and multiplicative family is clonal, and the principal ratio of any spectral extremal graph satisfies gamma=1+O(n^-1).
    Critical for Theorem 3.10(2)-(3); imported from [4, Theorem 4.6 and Lemma 3.6] without proof.
  • domain assumption Degree stability of F^(r) for (l+1)-color critical F with respect to Col(K_l^r), as stated in Table 1 of [12].
    Load-bearing input for Theorem 4.8 and Corollary 5.4; not proved in this paper.
  • domain assumption Members of the forbidden family F contain no isolated vertices (stated at the start of Section 4).
    Needed so expansion graphs and degree-stability thresholds behave as written.
  • domain assumption Lemma 4.6 from [18]: among l-partite r-graphs, the balanced complete l-partite hypergraph has strictly largest alpha-spectral radius.
    Converts P-colorability to the explicit extremal T_l^r(n) in Corollary 4.7 and Theorem 4.8.

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Pith. "Pith review of Spectral Tur\'an-type problems for the $\alpha$-spectral radius of hypergraphs with degree stability." pith.science (2026). https://pith.science/paper/FRFBLHQL

@misc{pith2026250924354,
  author       = {Pith},
  title        = {Pith review of: Spectral Tur\'an-type problems for the $\alpha$-spectral radius of hypergraphs with degree stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FRFBLHQL}},
  note         = {Machine review of arXiv:2509.24354}
}
abstract

An $r$-pattern $P$ is an ordered pair $P=([l],E)$, where $l$ is a positive integer and $E$ is a set of $r$-multisets with elements from $[l]$. An $r$-graph $H$ is said to be $P$-colorable if there is a homomorphism $\phi$: $V(H)\rightarrow [l]$ such that $\{\phi(v_{1}),\ldots,\phi(v_{r})\}\in E$ for every edge $\{v_{1},\ldots,v_{r}\}\in E(H)$. Let $\mathrm{Col}(P)$ denote the family of all $P$-colorable $r$-graphs. This paper studies spectral extremal problems for $\alpha$-spectral radius of hypergraphs via analytic techniques. We first prove that for any $r$-pattern $P$, the hypergraph attaining the maximum $\alpha$-spectral radius in $\mathrm{Col}(P)$ is asymptotically regular. Specifically, we establish asymptotically tight lower bounds for the minimum component of the principal eigenvector and the minimum degree of the spectral extremal hypergraphs in $\mathrm{Col}(P)$. Building on this regularity, we further show that for any family $\mathcal{F}$ of $r$-graphs that is degree-stable with respect to $\mathrm{Col}(P)$, spectral Tur\'an-type problems can be completely reduced to spectral extremal problems within $\mathrm{Col}(P)$. As an application, we determine the maximum $\alpha$-spectral radius ($\alpha\geq1$) among all $n$-vertex $F^{(r)}$-free $r$-graphs, where $F^{(r)}$ is the $r$-expansion of the color-critical graph $F$. This provides a powerful reduction tool for handling spectral Tur\'{a}n-type problems in hypergraphs. Finally, leveraging the spectral method, we derive a corresponding edge Tur\'an extremal result. More precisely, we show that if $\mathcal{F}$ is degree-stable with respect to $\mathrm{Col}(P)$, then every $\mathcal{F}$-free edge extremal hypergraph must be a $P$-colorable hypergraph.

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Forward citations

Cited by 1 Pith paper

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

  1. Spectral Radius Conditions for 3-Uniform Intersecting Families

    math.CO 2026-07 unverdicted novelty 5.0 of 10

    For sufficiently large n, the maximum spectral radii of M_{k+1}-free and non-trivial intersecting 3-graphs on n vertices are determined and the extremal hypergraphs are characterized.

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    , xn/n1−1/α))

    Then λ(α)(G) nr−r/α ≤ PG(x) nr−r/α =P G((x1/n1−1/α, . . . , xn/n1−1/α)). By Power-Mean inequality, for anyα≥1, x1 +· · ·+xn n1−1/α ≤(x α 1 +· · ·+xα n)1/α = 1. From Lemma 2.2 it follows that λ(α)(G) nr−r/α ≤λ (1)(G)≤λ (1)(P, n)≤λ (1)(P) =π(P). Sinceλ (α)(G)≥r!e(G)/n r/α, we co...

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Reviewed August 4, 2026 · model on record in the stance chip above.