Spectral Turán-type problems for the $α$-spectral radius of hypergraphs with degree stability

An r-pattern P is defined as 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 $ϕ$: V(H)→ [l] such that the r-multiset \ϕ(v₁),…,ϕ(vᵣ)\ is in E for every edge ₁,…,vᵣ\∈ E(H). Let $Col(P)$ denote the family of all P-colorable r-graphs. This paper establishes spectral extremal results for $α$-spectral radius of hypergraphs using analytic techniques. We show that for any family F of r-graphs that is degree-stable with respect to $Col(P)$, spectral Turán-type problems can be effectively reduced to spectral extremal problems within $Col(P)$. As an application, we determine the maximum $α$-spectral radius (α≥1) among all n-vertex F⁽ʳ⁾-free r-graphs, where F⁽ʳ⁾ represents the r-expansion of the color critical graph F. We also characterize the corresponding extremal hypergraphs. Furthermore, leveraging the spectral method, we derive a corresponding edge Turán extremal result. More precisely, we show that if F is degree-stable with respect to $Col(P)$, then every F-free edge extremal hypergraph must be a P-colorable hypergraph.