On the Distances between Latin Squares and the Smallest Defining Set Size

Abstract

In this note we show that for each Latin square L of order n ≥ 2, there exists a Latin square L ′ = L of order n such that L and L ′ differ in at most 8 √ n cells. Equivalently, each Latin square of order n contains a Latin trade of size at most 8 √ n. We also show that the size of the smallest defining set in a Latin square is Ω(n 3/2 ).

Teaching dimension, VC dimension, and critical sets in Latin squares

“…Cavenagh [Cav07] gave the first superlinear lower-bound of n⌊(log n) 1/3 /2⌋ in 2007. Recently, Cavenagh and Ramadurai [CR16] improved this bound to Ω(n 3/2 ).…”