The Helly number of Hamming balls and related problems

TL;DR

This paper establishes a Helly-type theorem for Hamming balls with bounded radius, providing tight bounds and a novel dimension-independent proof technique, with implications for extremal set theory and graph theory.

Contribution

It introduces a new Helly theorem for Hamming balls with a dimension-independent proof method, extending classical results to discrete metric spaces.

Findings

Helly's theorem variant proven for Hamming balls with radius t

The result is tight for all |X|>1 and n>t

New dimension argument technique developed

Abstract

We prove the following variant of Helly's classical theorem for Hamming balls with a bounded radius. For $n>t$ and any (finite or infinite) set $X$, if in a family of Hamming balls of radius $t$ in $X^n$, every subfamily of at most $2^{t+1}$ balls have a common point, so do all members of the family. This is tight for all $|X|>1$ and all $n>t$. The proof of the main result is based on a novel variant of the so-called dimension argument, which allows one to prove upper bounds that do not depend on the dimension of the ambient space. We also discuss several related questions and connections to problems and results in extremal finite set theory and graph theory.