Subsquares in Random Latin Squares and Rectangles

ABSTRACT A partial Latin rectangle is if the number of nonempty entries in each row and column is at most and each symbol is used at most times. We prove that the probability a uniformly random Latin rectangle, where , contains a ‐sparse partial Latin rectangle with nonempty entries is for sufficiently large and sufficiently small . Using this result, we prove that a uniformly random order‐ Latin square asymptotically almost surely has no Latin subsquare of order greater than for an absolute constant .