Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T ) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application we prove that non-partite Ramanujan complexes have 'high girth' and high chromatic number, generalizing a well known result about Ramanujan graphs.
In [Ho] A.J. Hoffman proved a lower bound on the chromatic number of a graph in the terms of the largest and the smallest eigenvalues of its adjacency matrix. In this paper, we prove a higher dimensional version of this result and give a lower bound on the chromatic number of a pure d-dimensional simplicial complex in the terms of the spectra of the higher Laplacian operators.
The n-th tensor power of a graph with vertex set V is the graph on the vertex set V n , where two vertices are connected by an edge if they are connected in each coordinate. The problem of studying independent sets in tensor powers of graphs is central in combinatorics, and its study involves a beautiful combination of analytical and combinatorial techniques. 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 Turán 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 following problem of Frankl from 1990. An extended triangle in a family of sets is a triplet {A, B, C} ⊆ [n] 2k such that each element of [n] belongs either to none of the sets {A, B, C} or to exactly two of them. Frankl asked how large can a family F ⊆ [n] 2k be if it does not contain a triangle. We show that if 1 2 n ≤ 2k ≤ 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.
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