Spectral generalized Turán problems

Combining two well-studied variants of the classical Turán problem, the generalized Turán problem and the spectral Turán problem, we introduce the spectral generalized Turán problem and establish a general theorem that extends the result of Keevash–Lenz–Mubayi~\cite{KLM14} on the spectral Turán problem in this broader setting. As a quick application, we obtain the spectral Erdős Pentagon Theorem. We also introduce the notion of entropic density for generalized Turán problems, and show that it coincides with the generalized spectral radius, extending a recent result of Chao–Hans on entropic Turán density.