The Helly number of Hamming balls and related problems

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.