Subsquares in random Latin squares and rectangles

A $k\times n$ partial Latin rectangle is $C\mathit{\text{-sparse}}$ if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k\times n$ Latin rectangle, where $k<(1\mathrm{/}2-\alpha )n$, contains a $\beta n$-sparse partial Latin rectangle with $\mathrm{\ell }$ nonempty entries is $(\frac{1\pm \epsilon }{n}{)}^{\mathrm{\ell }}$ for sufficiently large $n$ and sufficiently small $\beta$. Using this result, we prove that a uniformly random order-$n$ Latin square asymptotically almost surely has no Latin subsquare of order greater than $c\sqrt{n\log n}$ for an absolute constant $c$.