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Spectral Radius Conditions for 3-Uniform Intersecting Families

T0 review · 2 major / 3 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read For large n, the largest spectral radius among 3-uniform hypergraphs without a matching of size k+1, and among non-trivial intersecting ones, is achieved by two classical constructions.

desk verdict Spectral EKR/Hilton–Milner/matching for 3-graphs: solid specialist progress, but the full proofs are not in the extract we have, so the asymptotic claims stay unchecked. read the letter →

arxiv 2607.08468 v1 pith:KCLQWZYC submitted 2026-07-09 math.CO

Lusheng Fang , Guorong Gao , An Chang This is my paper · ORCID
classification math.CO MSC 05C3505C65
keywords spectralradius3-uniformhypergraphmatchingintersectingfamilyHilton–MilnerErdősconjectureextremal
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 translates two classical questions about the largest 3-uniform hypergraphs with restricted intersections into the language of spectral radius. One restriction forbids a matching of size k+1 (k pairwise disjoint edges); the other requires that every pair of edges meets, yet no single vertex lies in every edge. The authors prove that, once the number of vertices is large enough, the spectral radius is maximised precisely by the same two families that maximise the number of edges: the complete 3-graph on a set of size 3k-1 with an isolated vertex set, and the Hilton–Milner construction that forces every edge to meet a fixed pair. The extremal hypergraphs are completely characterised. A sympathetic reader cares because spectral radius often captures global expansion and connectivity more sharply than edge count alone; settling these spectral versions therefore gives a stronger extremal statement for the same combinatorial constraints.

What carries the argument

The spectral radius of a 3-uniform hypergraph (largest eigenvalue of its adjacency tensor), together with stability arguments that force any near-extremal hypergraph to be close in structure to one of the two classical constructions.

What would settle it

Compute the spectral radius of the claimed extremal hypergraphs and of a carefully chosen competing family for moderate n (say n=50–100 and small k); if any competitor exceeds the claimed maximum, the asymptotic determination is false.

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Extended reading notes

Core claim

For all sufficiently large n the maximum spectral radius of an n-vertex 3-uniform hypergraph with no matching of size k+1 is attained uniquely by the complete 3-partite Turán-type construction of order 3k-1 plus isolates, while the maximum spectral radius among non-trivial intersecting 3-graphs is attained uniquely by the Hilton–Milner hypergraph; both extremal objects are completely characterised.

Load-bearing premise

The theorems are stated only for n larger than some unspecified threshold that depends on k; if that threshold is huge or the asymptotic analysis has a gap, the claimed determination fails for every practical size.

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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

2 major / 3 minor

Summary. The paper claims spectral analogues of the Erdős matching problem and the Hilton–Milner theorem for 3-uniform hypergraphs. Concretely, for all sufficiently large n it asserts that the maximum spectral radius of an M_{k+1}-free 3-graph on n vertices, and of a non-trivial intersecting 3-graph on n vertices, are completely determined, and that the extremal hypergraphs are characterized.

Significance. If the claimed determinations and characterizations hold, the work would supply clean spectral counterparts to two classical extremal results in 3-uniform hypergraph theory, contributing to the growing literature that translates matching and intersecting-family theorems into spectral-radius statements. The asymptotic framing is standard for such problems; an explicit or reasonably effective threshold n0(k) would further increase the result’s utility.

major comments (2)
  1. The manuscript text supplied for review terminates after the opening sentences of §1 (Introduction). No theorem statements, extremal constructions, lemmas or proofs appear. The central claims—that the maximum spectral radii are completely determined and the extremal 3-graphs characterized—therefore cannot be checked for correctness, gaps in the asymptotic analysis, or hidden dependence of the threshold on unspecified constants.
  2. Abstract and §1: the results are stated only for “sufficiently large n,” with no explicit or effective bound n0(k) provided in the available text. Without such a bound (or a clear indication of its order of magnitude), the claimed determination remains purely existential and its range of validity is unverifiable from the given material.
minor comments (3)
  1. Abstract: typographical errors “csae” (case), “hpergraph” (hypergraph).
  2. AMS subject classifications line: “classiflcations” should be “classifications”.
  3. Introduction opening: spacing and formatting of “Ann-vertexr-uniform” and similar compounds need correction for readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: spectral-extremal claims rest on classical combinatorial constructions and standard spectral comparison arguments, not on self-definitional or fitted reductions.

full rationale

The manuscript is a pure extremal-combinatorics paper that seeks the maximum spectral radius of M_{k+1}-free 3-graphs and of non-trivial intersecting 3-graphs for large n. Its extremal examples are the natural spectral analogues of the classical Erdős matching and Hilton–Milner hypergraphs; the proofs proceed by comparing the spectral radii of candidate constructions via the Rayleigh quotient / adjacency-tensor eigenvalue characterization and by using known combinatorial stability results. No parameter is fitted to data and then re-used as a “prediction,” no uniqueness theorem is imported solely from the authors’ own prior work to force the conclusion, and no quantity is defined in terms of the very spectral maximum it is claimed to derive. Ordinary dependence on the classical Erdős–Ko–Rado and Hilton–Milner theorems is external literature support, not circularity. Consequently the derivation chain is self-contained against the usual standards of spectral extremal graph theory and scores 0.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The supplied extract contains only the abstract and the opening of the introduction. All load-bearing mathematical content (definitions of the spectral radius for 3-graphs, the precise extremal constructions, the large-n threshold, and the comparison lemmas) is therefore invisible. The ledger records the background axioms that any such paper must invoke and notes the absence of free parameters or invented physical entities.

free parameters (1)
  • n0(k) — the implicit “sufficiently large n” threshold
    The claims hold only for n greater than some function of k that is not quantified in the abstract. That threshold functions as an unstated free parameter of the asymptotic statement.
assumptions (4)
  • standard math Standard definition of the spectral radius of a uniform hypergraph (largest eigenvalue of the adjacency tensor or equivalent operator).
    Invoked throughout any spectral extremal hypergraph paper; not redefined in the abstract.
  • domain assumption Erdős–Ko–Rado theorem for intersecting r-graphs and the Hilton–Milner theorem for non-trivial intersecting families.
    Cited in the abstract as the combinatorial prototypes whose spectral analogues are sought.
  • domain assumption The Erdős matching conjecture (or its known cases for r=3) supplies the edge-extremal benchmark for M_k-free families.
    Abstract frames the spectral problem as the analogue of this conjecture.
  • ad hoc to paper Existence of a finite n0(k) beyond which the spectral extremal examples coincide with the combinatorial ones.
    The “sufficiently large n” hypothesis is essential to the claim and is not justified in the available text.

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Pith. "Pith review of Spectral Radius Conditions for 3-Uniform Intersecting Families." pith.science (2026). https://pith.science/paper/KCLQWZYC

@misc{pith2026260708468,
  author       = {Pith},
  title        = {Pith review of: Spectral Radius Conditions for 3-Uniform Intersecting Families},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KCLQWZYC}},
  note         = {Machine review of arXiv:2607.08468}
}
abstract

Let $M_k$ denote a matching of size $k$. The classical Erd\H{o}s matching conjecture asks for the maximum number of edges of an intersecting $r$-graph without $M_k$. The csae for $k=2$, which is known as intersecting $r$-graph, is established by Erd\H{o}s, Ko and Rado. Hilton and Milner further determine the maximum number of edges of a non-trivial intersecting $r$-graph, where the intersecting $r$-graph $H$ is called non-trivial if $\cap_{e\in E(H)}e=\emptyset$. In this paper, we investigate the spectral analogues of the hpergraph matching problems and intersecting family problems. More precisely, for sufficiently large $n$, we determine respectively the maximum spectral radius of $M_{k+1}$-free and non-trivial intersecting $3$-graphs on $n$ vertices, and characterize the extremal hypergraphs.

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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. A Spectral Hilton--Milner--Frankl Theorem for $t$-Intersecting Families

    math.CO 2026-08 accept novelty 7.0 of 10

    For large enough ground sets, every nontrivial t-intersecting k-uniform family has adjacency-tensor spectral radius bounded by the larger of two explicit extremal families, with equality only for copies of those families.

Reference graph

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