The Helly theorem for Hamming balls and related problems

Helly's theorem, proved more than a century ago, is a fundamental result in discrete geometry. It states that if every d+1 sets in a finite family of convex sets in d-dimensional Euclidean space has a nonempty intersection, then the entire family has a nonempty intersection.

In this talk, we present a version of Helly's theorem for Hamming balls with bounded radius. Our proof is based on a novel variant of the so-called dimension argument, which enables us to establish upper bounds that are independent of the dimension of the ambient space. We also discuss several connections between our result and problems in extremal set theory, coding theory, and graph theory.

Joint work with Noga Alon and Zhihan Jin.