There are no good infinite families of toric codes

Soprunov and Soprunova posed a question on the existence of infinite families of toric codes that are “good” in a precise sense. We prove that such good families do not exist by proving a more general Szemerédi-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.