On the threshold problem for Latin boxes. Issue 4 (30th April 2019) - Luria, Zur
Abstract : 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 .