A recursive theta body for hypergraphs

TL;DR

This paper extends the Lovász theta body concept from graphs to hypergraphs recursively, exploring its properties and applications to combinatorial bounds and hypergraph problems.

Contribution

It introduces a recursive extension of the theta body for hypergraphs, linking it to high-dimensional Hoffman bounds and applying it to classical combinatorial problems.

Findings

Extended theta body for hypergraphs with fundamental properties

Relation to high-dimensional Hoffman bound

Applications to triangle-free graphs and Hamming cube sets

Abstract

The theta body of a graph, introduced by Gr\"otschel, Lov\'asz, and Schrijver in 1986, is a tractable relaxation of the independent-set polytope derived from the Lov\'asz theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the high-dimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: triangle-free graphs and Mantel's theorem, and bounds on the density of triangle-avoiding sets in the Hamming cube.