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Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number

T0 review · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Near-maximal tensor spectral radius forces a k-uniform hypergraph with bounded matching number to sit inside the star-like extremal example S_{n,k,β} and miss only O(δ n^{k-1}) of its edges.

desk verdict New spectral stability theorem for bounded-matching hypergraphs; clean eigenvector proof that also recovers the known exact extremal result for large n. read the letter →

arxiv 2607.23560 v1 pith:NBXDC6KI submitted 2026-07-26 math.CO

Yi Xu , Yi-Zheng Fan This is my paper · ORCID
classification math.CO MSC 05C6505C3515A69
keywords uniformhypergraphadjacencytensorspectralradiusstabilitymatchingnumberErdősconjecture
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 proves a stability theorem for the tensor spectral radius of k-uniform hypergraphs whose matching number is at most a fixed β. If an n-vertex example has spectral radius within a small factor of the known maximum, then there is a set of exactly β vertices that meets every edge, and the hypergraph differs from the classical extremal construction S_{n,k,β} by at most a constant times δ n^{k-1} edges. The argument works by reading the principal eigenvector: vertices with large coordinates must form a vertex cover of size β, after which a quantitative comparison of Rayleigh quotients bounds the missing edges. As a direct corollary one recovers the exact spectral Erdős matching theorem for all large n, without shifting. A sympathetic reader cares because spectral radius is often easier to compute or bound than edge counts, yet here it still forces the same rigid structure that the classical extremal problem predicts.

What carries the argument

The normalized nonnegative principal eigenvector of the adjacency tensor. Vertices whose coordinates exceed a small threshold η form a set W of size exactly β that covers every edge; the same vector then supplies the coefficient gap that converts a spectral deficit into an edge-deficit bound.

What would settle it

Exhibit a single infinite family of k-graphs with matching number ≤β whose spectral radius is (1-o(1)) times that of S_{n,k,β} yet whose edge sets remain a positive-density distance away from every copy of S_{n,k,β}.

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Extended reading notes

Core claim

For fixed k≥3 and β≥2 and all sufficiently large n, every n-vertex k-graph H with matching number at most β and tensor spectral radius at least (1-δ)ρ(S_{n,k,β}) admits a β-set W that intersects every edge of H; after relabeling, the symmetric difference of the edge sets of H and S_{n,k,β} has size at most Cδ n^{k-1}.

Load-bearing premise

n must be large enough that all lower-order error terms stay strictly smaller than the main-term gap between β and β-1 in the spectral-radius asymptotics.

Editorial extensions

If this is right

  • The exact spectral Erdős matching conjecture holds for all sufficiently large n: ρ(H)≤ρ(S_{n,k,β}) with equality only for H isomorphic to S_{n,k,β}.
  • Any hypergraph attaining spectral radius within o(n^{(k-1)^2/k}) of the extremal value must already be a spanning subgraph of some S_{n,k,β}.
  • The missing-edge count is linearly controlled by the spectral deficit, giving a quantitative stability form rather than a mere qualitative one.
  • The eigenvector method supplies an alternative to shifting that may extend to other bounded-matching spectral problems.

Reading between the lines

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

  • The same eigenvector-threshold argument should adapt to other hereditary properties whose extremal examples are complete multipartite or starring constructions.
  • Once n is large, the stability constant C can be tracked explicitly in terms of k and β, opening the door to effective (computer-checkable) bounds for moderate n.
  • Combining the stability theorem with existing edge-stability results for the Erdős matching conjecture would yield a two-way dictionary between spectral and combinatorial closeness.
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Editorial analysis

A structured set of objections, weighed in public.

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

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: self-contained variational proof from eigenvector equations and elementary inequalities

full rationale

The derivation chain is independent and non-circular. Theorem 1.1 is proved in two steps (Lemmas 3.1–3.2) directly from the variational characterization of the tensor spectral radius (Lemma 2.2 / Qi), the eigenvalue equation for a normalized nonnegative principal eigenvector, Hölder/Maclaurin inequalities, and the matching-number hypothesis μ(H)≤β. Lemma 2.3 supplies crude but self-contained bounds on ρ(S_{n,k,β}). Corollary 1.2 is obtained from Theorem 1.1 by setting δ=0; it does not import the concurrent Kang–Lu–Yuan–Zhou extremal theorem as a premise. Citations to prior Fan co-authored work ([21], [25]) and to [12] are background or comparison only and are not load-bearing for any coefficient, uniqueness claim, or structural conclusion. There is no fitted parameter, no self-definitional loop, and no uniqueness theorem smuggled from the authors’ own prior papers. The argument is a standard first-principles stability proof in spectral extremal hypergraph theory.

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

The paper is pure asymptotic extremal combinatorics. It rests on standard tensor spectral theory and classical inequalities; the only paper-specific ingredients are existential constant choices (δ₀,η,δ₁,n₀,C) that make coefficient gaps dominate lower-order terms. No empirical free parameters and no newly postulated physical or combinatorial entities.

assumptions (5)
  • standard math Perron–Frobenius theorem for nonnegative (weakly irreducible) tensors: spectral radius is an eigenvalue with a nonnegative (positive) eigenvector.
    Invoked in Section 2 to guarantee a normalized nonnegative principal eigenvector x used throughout Lemmas 3.1–3.2.
  • standard math Variational characterization ρ(A)=max{x^T A x^{k-1}: x≥0, ||x||_k=1} for nonnegative symmetric tensors (Qi).
    Lemma 2.2; used to write ρ(H)=k ∑_e x_e and to compare trial vectors against S_{n,k,β}.
  • standard math Hölder, Maclaurin, and power-mean inequalities for nonnegative vectors.
    Applied repeatedly in Lemmas 2.3, 3.1, and 3.2 to bound multilinear forms by ℓ_k-norms.
  • domain assumption Adjacency tensor of a k-uniform hypergraph as defined by Cooper–Dutle, with spectral radius equal to that of the tensor.
    Section 2 definition; the entire spectral theory of the paper is built on this convention.
  • ad hoc to paper For fixed k,β there exist δ₀,η,δ₁,n₀ such that all o(1) and O(n^{...}) error terms are absorbed by the main-term coefficient gap.
    Explicit constant choices (5), the sentence after (9), and the choice of δ₁ before (12); load-bearing for both |W|=β and the edit-distance bound.

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Pith. "Pith review of Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number." pith.science (2026). https://pith.science/paper/NBXDC6KI

@misc{pith2026260723560,
  author       = {Pith},
  title        = {Pith review of: Tensor Spectral Stability for Uniform Hypergraphs with Bounded Matching Number},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NBXDC6KI}},
  note         = {Machine review of arXiv:2607.23560}
}
abstract

We establish a tensor spectral stability theorem for uniform hypergraphs with bounded matching number. More precisely, for fixed integers $k\geq 3$ and $\beta\geq2$, and sufficiently large $n$, we prove that every $n$-vertex $k$-uniform hypergraph $H$ with matching number at most $\beta$ and tensor spectral radius close to the maximum possible value among all such hypergraphs must be structurally close to the extremal hypergraph $S_{n,k,\beta}$, whose edges consist of all $k$-sets intersecting a fixed set of $\beta$ vertices. Furthermore, we show that every edge of $H$ intersects this distinguished vertex set and that $H$ contains all but a small proportion of the edges of $S_{n,k,\beta}$. As an application, we obtain a new proof of the spectral version of the Erd\H{o}s matching conjecture for sufficiently large $n$.

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