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 alpha-spectral radius defined using the alpha-norm for any alpha > 1; the usual spectral radius is the case alpha = 2. Our results imply that any hypergraph Tur'an 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 color-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 a-spectral version of the Erdos-Ko-Rado theorem on t-intersecting k-uniform hypergraphs.