Spectral Tur\'an problems for nondegenerate hypergraphs

TL;DR

This paper extends spectral stability results for nondegenerate hypergraphs, enabling the derivation of spectral Turán theorems for hypergraphs with bipartite or multipartite patterns, and characterizes extremal hypergraphs for various classes.

Contribution

It establishes a spectral stability theorem for nondegenerate hypergraphs, broadening the Keevash–Lenz–Mubayi criterion and linking spectral extremal problems to combinatorial stability.

Findings

Derived spectral Turán theorems for bipartite and multipartite hypergraphs.

Determined maximum spectral radius for several hypergraph classes.

Characterized extremal hypergraphs such as expansions, fans, and triangles.

Abstract

Keevash, Lenz and Mubayi developed a general criterion for hypergraph spectral extremal problems in their seminal work (SIAM J. Discrete Math., 2014). Their framework shows that extremal results on the $\alpha$-spectral radius (for $\alpha > 1$) may be deduced from a corresponding hypergraph Tur\'an problem exhibiting stability properties, provided its extremal construction satisfies certain continuity assumptions. In this paper, we establish a spectral stability result for nondegenerate hypergraphs, extending the Keevash--Lenz--Mubayi criterion. Applying this result, we derive two general spectral Tur\'an theorems for hypergraphs with bipartite or multipartite pattern, thereby transforming spectral Tur\'an problems into the corresponding purely combinatorial problems related to degree-stability in nondegenerate $k$-graph families. As applications, we determine the maximum $\alpha$-spectral radius for several classes of hypergraph and characterize the corresponding extremal hypergraphs, such as the expansion of complete graphs, the generalized fans, the cancellative hypergraphs, the generalized triangles, and a special book hypergraph.