  One powerful method for upper-bounding the largest independent set in a graph
is the Hoffman bound, which gives an upper bound on the largest independent set
of a graph in terms of its eigenvalues. It is easily seen that the Hoffman
bound is sharp on the tensor power of a graph whenever it is sharp for the
original graph.
  In this paper, we introduce the related problem of upper-bounding independent
sets in tensor powers of hypergraphs. We show that many of the prominent open
problems in extremal combinatorics, such as the Tur\'an problem for
(hyper-)graphs, can be encoded as special cases of this problem. We also give a
new generalization of the Hoffman bound for hypergraphs which is sharp for the
tensor power of a hypergraph whenever it is sharp for the original hypergraph.
  As an application of our Hoffman bound, we make progress on the problem of
Frankl on families of sets without extended triangles from 1990. We show that
if $\frac{1}{2}n\le2k\le\frac{2}{3}n,$ then the extremal family is the star,
i.e. the family of all sets that contains a given element. This covers the
entire range in which the star is extremal. As another application, we provide
spectral proofs for Mantel's theorem on triangle-free graphs and for
Frankl-Tokushige theorem on $k$-wise intersecting families.
