Spectral extremal problems for hypergraphs

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 α-spectral radius 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 generalizes a theorem of Nikiforov who proved stronger results in the case α=2. We also obtain an α-spectral version of the Erdős-Ko-Rado theorem on t-intersecting k-uniform hypergraphs.

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