Latin squares without proper subsquares

TL;DR

This paper proves the existence of high-dimensional Latin hypercubes without smaller Latin subhypercubes for most orders and dimensions, resolving a long-standing conjecture in combinatorics.

Contribution

It establishes the existence of Latin hypercubes without proper subhypercubes for all but two small cases, solving Hilton's 50-year-old conjecture for Latin squares.

Findings

Existence of such hypercubes for all but (4,2) and (6,2)

Resolution of Hilton's conjecture for Latin squares

Construction methods for Latin hypercubes without proper substructures

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 $d \geq 2$ 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.