Subsquares in Random Latin Rectangles

Suppose that k is a function of n and . We show that with probability \(1-O(1/n)\), a uniformly random \(k\times n\) Latin rectangle contains no proper Latin subsquare of order 4 or more, proving a conjecture of Divoux, Kelly, Kennedy and Sidhu. We also show that the expected number of subsquares of order 3 is bounded and find that the expected number of subsquares of order 2 is \(\left( {\begin{array}{c}k\\ 2\end{array}}\right) (1/2+o(1))\) for all \(k\leqslant n\).