On the threshold problem for Latin boxes

Let m  ≤  n  ≤  k . An m  ×  n  ×  k 0‐1 array is a Latin box if it contains exactly m n ones, and has at most one 1 in each line. As a special case, Latin boxes in which m  =  n  =  k are equivalent to Latin squares. Let M ( m , n , k ; p ) be the distribution on m  ×  n  ×  k 0‐1 arrays where each entry is 1 with probability p , independently of the other entries. The threshold question for Latin squares asks when M ( n , n , n ; p ) contains a Latin square with high probability. More generally, when does M ( m , n , k ; p ) support a Latin box with high probability? Let ε  > 0. We give an asymptotically tight answer to this question in the special cases where n  =  k and m ≤ 1 − ε n , and where n  =  m and k ≥ 1 + ε n . In both cases, the threshold probability is Θ log n / n . This implies threshold results for Latin rectangles and proper edge‐colorings of K n , n .