Intercalates in double and triple arrays

Abstract

AbstractThis paper addresses the question of how intercalates occur in the two known infinite families of triple arrays, the Paley triple arrays constructed in 2005 by Preece et al., and the Triple arrays from difference sets in 2017 by Nilson and Cameron. The main reason for doing this is that the number of such embedded Latin squares is often used when checking whether two arrays are isotopic or not. We determine sharp bounds for the number and density of intercalates for the main subclasses of these families respectively. We also prove the existence of an infinite family of triple arrays in which every two occurrences of an entry lie in an intercalate.