Latin squares without proper subsquares

2023/10/03 by Allsop, Jack, Wanless, Ian M. · 3 citations
#05B15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.01923

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) ∉ \(4,2), (6,2)\ with d ≥ 2 there exists a d-dimensional Latin hypercube of order n that contains no d-dimensional Latin subhypercube of any order in \2,…,n-1\. The d=2 case settles a 50 year old conjecture by Hilton on the existence of Latin squares without proper subsquares.

Cited by

• Further Results on Latin Squares with Disjoint Subsquares using Rational Outline Squares
• Erdős meets Nash-Williams
• Perfect 1-factorisations of K_11,11

Related

• Substructures in Latin squares
• Latin cubes with disjoint subcubes of two orders
• Latin Squares whose transversals share many entries
• Latin squares with three disjoint subsquares of the same order
• Latin Squares whose transversals intersect in unusual ways