  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.

Combinatorics