High dimensional Hoffman bound and applications in extremal combinatorics

TL;DR

This paper extends the Hoffman bound to hypergraphs, providing a new tool for extremal combinatorics problems, and applies it to solve classical problems like the Turán problem, Frankl's set families, and spectral proofs of Mantel's theorem.

Contribution

It introduces a generalized Hoffman bound for hypergraphs that is sharp for tensor powers, and connects it to key extremal combinatorics problems.

Findings

New hypergraph Hoffman bound generalization

Progress on Frankl's problem for set families

Spectral proofs of classical combinatorics theorems

Abstract

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.