High dimensional Hoffman bound and applications in extremal combinatorics

The n-th tensor power of a graph with vertex set V is the on n , where two vertices are connected by an edge if they in each coordinate. One powerful method for upper-bounding largest independent Hoffman bound, which gives upper bound terms its eigenvalues. In this paper we introduce problem sets powers hypergraphs. We show that many prominent open problems extremal combinatorics, such as Turán graphs and hypergraphs, can be encoded special cases problem. generalize to give several applications.