There are no good infinite families of toric codes

Abstract: Soprunov and Soprunova introduced the notion of a good infinite family of toric codes. We prove that such good families do not exist by proving a more general Szemer\'edi-type result: for all $c\in(0,1]$ and all positive integers $N$, subsets of density at least $c$ in ${0,1,\dots,N-1}^n$ contain hypercubes of arbitrarily large dimension as $n$ grows.