Spectral extremal problems for hypergraphs

Abstract: In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from `strong stability' forms of the corresponding (pure) extremal results. These results hold for the \alpha-spectral radius defined using the \alpha-norm for any \alpha>1; the usual spectrum is the case \alpha=2. Our results imply that any hypergraph Tur\'{a}n problem which has the stability property and whose extremal construction satisfies some rather mild continuity assumptions admits a corresponding spectral result. A particular example is to determine the maximum \alpha-spectral radius of any 3-uniform hypergraph on n vertices not containing the Fano plane, when n is sufficiently large. Another is to determine the maximum \alpha-spectral radius of any graph on n vertices not containing some fixed colour-critical graph, when n is sufficiently large; this generalizes a theorem of Nikiforov who proved stronger results in the case \alpha=2. We also obtain an \alpha-spectral version of the Erd\H{o}s-Ko-Rado theorem on t-intersecting k-uniform hypergraphs.