An efficient container lemma

TL;DR

This paper introduces a refined hypergraph container theorem tailored for hypergraphs with large uniformities, leveraging convex geometry to produce smaller container families and improve bounds in various combinatorial problems.

Contribution

It presents a new, efficient container theorem for large uniformity hypergraphs using convex geometry, enhancing bounds in extremal combinatorics and related fields.

Findings

Smaller families of containers for independent sets in large uniformity hypergraphs.

Improved bounds in extremal graph theory, discrete geometry, and Ramsey theory.

Novel application of convex geometry in hypergraph container construction.

Abstract

We prove a new, efficient version of the hypergraph container theorems that is suited for hypergraphs with large uniformities. The main novelty is a refined approach to constructing containers that employs simple ideas from high-dimensional convex geometry. The existence of smaller families of containers for independent sets in such hypergraphs, which is guaranteed by the new theorem, allows us to improve upon the best currently known bounds for several problems in extremal graph theory, discrete geometry, and Ramsey theory.