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 spectral radius is the case $\alpha=2$. 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 $\alpha$-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 $\alpha$-spectral version of the Erdös--Ko--Rado theorem on $t$-intersecting $k$-uniform hypergraphs.