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 a generalised α-spectrum defined using the α-norm for any α> 1; the usual spectrum is the case α = 2. Our results imply that any hypergraph Turán 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 α-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 α-spectral radius of any graph on n vertices not containing some fixed colour-critical graph, when n is sufficiently large; this generalises a theorem of Nikiforov. We also obtain an α-spectral version of the Erdős-Ko-Rado theorem on t-intersecting k-uniform hypergraphs.
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