Latin squares without proper subsquares

Abstract

A $d$-dimensional Latin hypercube of order $n$ is a $d$-dimensional array containing symbols from a set of cardinality $n$ with the property that every axis-parallel line contains all $n$ symbols exactly once. We show that for $(n,d)\notin \{(4,2),(6,2)\}$ with $d\ge 2$ there exists a $d$-dimensional Latin hypercube of order $n$ that contains no $d$-dimensional Latin subhypercube of any order in $\{2,\dots ,n-1\}$. The $d=2$ case settles a 50 year old conjecture by Hilton on the existence of Latin squares without proper subsquares.