Subsquares in random Latin squares and rectangles

Abstract

A $k\times n$ partial Latin rectangle is \textit{$C$-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 $k<(1/2-\alpha )n$, contains a $\beta n$-sparse partial Latin rectangle with $\ell$ nonempty entries is $(\frac{1\pm \epsilon }{n}{)}^{\ell }$ 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 $c$.

11 pages; 1 page appendix, corrected a typo in random Steiner system conjecture