Spectral Radius Conditions for 3-Uniform Intersecting Families

Abstract: Let $M_k$ denote a matching of size $k$. The classical Erdős 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 Erdős, 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.

• The paper proves that, for sufficiently large n, A₁³ uniquely maximizes spectral radius among Mₖ₊₁-free 3-graphs when k=o(n), while A₃³ takes over when k is proportional to n/3.
• The paper shows that H₁ uniquely maximizes the spectral radius of non-trivial intersecting 3-graphs, with ρ(H₁)=∛12·n²ᐟ³⁺ᵒ⁽¹⁾ exceeding ρ(F₁³)=∛9·n²ᐟ³⁺ᵒ⁽¹⁾.
• The paper combines shifting, Perron eigenvector symmetry, candidate chains, and asymptotic spectral comparisons to establish these results while leaving intermediate k-regimes, finite thresholds, and uniformities r≥4 open.

This paper by Fang, Gao, and Chang addresses spectral extremal problems for 3-uniform hypergraphs whose edge sets are constrained by matching conditions or non-trivial intersection properties. The work extends two classical results in extremal combinatorics—the Erdős Matching Conjecture framework and the Hilton–Milner theorem—to the setting of adjacency tensor spectral radii.

Background and motivation

For an $r$-graph $H$, the spectral radius $\rho(H)$ is the largest absolute eigenvalue of its adjacency tensor $\mathcal{A}(H)$, defined via the Cooper–Dutle normalization. A central problem in spectral hypergraph theory, originating with Brualdi and Solheid's question for graphs, asks for the maximum spectral radius among $r$-graphs in a prescribed class. When the forbidden family is fixed, this is the spectral Turán problem. Prior work has concentrated largely on nondegenerate forbidden configurations (e.g., Fano-free 3-graphs by Keevash, Lenz, and Mubayi; cancellative 3-graphs by Ni, Liu, and Kang; expansions of color-critical graphs by Gao–Chang–Hou and She et al.), while degenerate cases—families with Turán density zero—remain comparatively underexplored.

The present paper targets two canonical degenerate settings. First, the Erdős Matching Conjecture asserts that an $M_{k+1}$-free $r$-graph on $n \geq (k+1)r$ vertices has at most $\max\{|\mathcal{A}^r_1|, |\mathcal{A}^r_r|\}$ edges, where $\mathcal{A}^r_i = \{e : |e \cap [ik+i-1]| \geq i\}$. Erdős proved that for large $H$0, the first candidate $H$1 is optimal in the counting sense. Second, the Hilton–Milner theorem determines the maximum size of a non-trivial intersecting $H$2-graph—one with empty total core—showing that $H$3 (and additionally $H$4 when $H$5) are the unique extremal families. The natural question is whether these same structures also maximize the spectral radius.

Main results

The paper establishes three spectral extremal theorems for sufficiently large $H$6:

Spectral Erdős matching theorem ($H$7). If $H$8 is an $H$9-free 3-graph on $\rho(H)$0 vertices with $\rho(H)$1 and $\rho(H)$2, then $\rho(H)$3, with equality attained uniquely at $\rho(H)$4. This confirms that the "all edges meeting a fixed $\rho(H)$5-set" construction is spectrally dominant in the sparse-matching regime.

Spectral Erdős matching theorem ($\rho(H)$6). In the dense regime where $\rho(H)$7 grows linearly as $\rho(H)$8, the answer switches to the other Erdős candidate: $\rho(H)$9, with equality at $\mathcal{A}(H)$0, i.e., the complete 3-graph on $\mathcal{A}(H)$1 vertices plus isolated vertices. This phase transition between the two extremal candidates mirrors the combinatorial picture and is arguably the paper's most substantive contribution, since it identifies the spectral analogue of the threshold behavior in the Erdős Matching Conjecture.

Spectral Hilton–Milner theorem. If $\mathcal{A}(H)$2 is a non-trivial intersecting 3-graph on $\mathcal{A}(H)$3 vertices, then $\mathcal{A}(H)$4, with equality at $\mathcal{A}(H)$5, where $\mathcal{A}(H)$6. Notably, the authors show asymptotically that

$\mathcal{A}(H)$7

so since $\mathcal{A}(H)$8, the $\mathcal{A}(H)$9 structure strictly dominates spectrally even though both are extremal in the Hilton–Milner counting sense. This is a genuine divergence between the counting and spectral notions: uniqueness of the spectral extremal configuration fails to inherit the non-uniqueness of the classical result.

Methodology

All three proofs follow a common architecture built on shifting. The Keough–Radcliffe shifting operation $r$0 preserves the $r$1-free property (proved here directly by an exchange argument on putative matchings) and does not decrease spectral radius, so an extremal hypergraph may be assumed shifted. In each case, the authors construct a finite chain of shifted candidate hypergraphs $r$2 avoiding prescribed disjoint edges, argue via the eigenvector equation and vertex-equivalence symmetries (Nikiforov's lemma forcing equal Perron coordinates on equivalent vertices) that the spectral radius increases strictly along the chain, and conclude that the terminal member—which coincides with the claimed extremal graph—is optimal.

The quantitative comparisons rest on asymptotic bookkeeping: writing $r$3 and $r$4 for the Perron coordinates on the partition classes, the authors combine the eigenvector equations with the crude bound $r$5 from Keevash–Lenz–Mubayi to pin down exponent relations, then verify that the leading-order terms in the spectral difference $r$6 are positive. For the Hilton–Milner proof, an additional structural obstacle arises: shifting can destroy non-triviality. The authors handle this by showing that if a shift trivializes the family, one may instead embed a complete 4-vertex subconfiguration $r$7 that is invariant under all subsequent shifts, thereby obtaining a shifted non-trivial intersecting extremal example. The final case analysis then eliminates all competing shifted structures by direct spectral comparison against both $r$8 and $r$9.

Limitations and open questions

Several restrictions qualify the results. All theorems hold only for sufficiently large $M_{k+1}$0, and no explicit threshold is given. The matching results cover only the regimes $M_{k+1}$1 and $M_{k+1}$2; the intermediate range where $M_{k+1}$3 grows linearly but with constant different from $M_{k+1}$4, as well as general uniformity $M_{k+1}$5, is left open. The proofs rely on asymptotic exponent comparisons rather than exact inequalities, so the method does not immediately yield finite-$M_{k+1}$6 statements. It would also be natural to ask whether the strict ordering $M_{k+1}$7 persists for small $M_{k+1}$8, and whether a full spectral version of the Erdős Matching Conjecture—with the correct extremal candidate across all ranges of $M_{k+1}$9 relative to $r$0—can be established for $r$1.

Conclusion

The paper resolves the spectral analogues of the Erdős matching problem and the Hilton–Milner problem for 3-graphs in the stated regimes, identifying $r$2, $r$3, and $r$4 respectively as the unique spectral extremal configurations for large $r$5. The identification of the phase transition between $r$6 and $r$7, together with the demonstration that $r$8 strictly dominates $r$9 spectrally despite their co-extremality in edge count, sharpens the correspondence between classical extremal set theory and spectral hypergraph theory while delineating precisely where that correspondence requires refinement.