This paper is part of the ongoing effort to study high-dimensional permutations. We prove the analogue to the Erdős-Szekeres theorem: For every k 1, every order-n k-dimensional permutation contains a monotone subsequence of length Ω k ( √ n), and this is tight. On the other hand, and unlike the classical case, the longest monotone subsequence in a random k-dimensional permutation of order n is asymptotically almost surely Θ k (n k/(k+1) ).

Let m ≤ n ≤ k. An m × n × k 0-1 array is a Latin box if it contains exactly mn 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, 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 (n, n, n; p) contains a Latin square with high probability. More generally, when does (m, n, k; p) support a Latin box with high probability? Let > 0. We give an asymptotically tight answer to this question in the special cases where n = k and m ≤ (1 − ) n, and where n = m and k ≥ (1 + ) 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 .

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Consider the random process in which the edges of a graph G are added one by one in a random order. A classical result states that if G is the complete graph K 2 n or the complete bipartite graph K n , n, then typically a perfect matching appears at the moment at which the last isolated vertex disappears. We extend this result to arbitrary k‐regular bipartite graphs G on 2 n vertices for all k = ω true( n log 1 / 3 n true). Surprisingly, this is not the case for smaller values of k. Using a construction due to Goel, Kapralov, and Khanna, we show that there exist bipartite k‐regular graphs in which the last isolated vertex disappears long before a perfect matching appears.