An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

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-nk-dimensional permutation contains a monotone subsequence of length Ωk($\sqrt{n}$), 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 Θk(nk/(k+1)).

A Latin square is reduced if its first row and first column are in natural order. For Latin squares of a particular order $n$, there are four possible different parities. We confirm a conjecture of Stones and Wanless by showing asymptotic equality between the numbers of reduced Latin squares of each possible parity as the order $n\rightarrow \infty$.

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) × (2k + 1) array A on the symbols (1,. . .,2k+1) with at most c(2k+1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1) × (2k+1) Latin square S on the symbols 1,. . .,2k+1 such that, for each i,j ∈ {1,. . .,2k+1}, the symbol in position (i,j) of S does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist. Using this result, we also show that there is a constant ρ, such that, for any positive integer n, if each cell in an n × n array B is assigned a set of m ≤ ρ n symbols, where each set is chosen independently and uniformly at random from {1,. . .,n}, then the probability that B is avoidable tends to 1 as n → ∞.

For a class of ‘linear’ sudoku solutions, we construct mutually orthogonal families of maximal size for all square orders, and we show that all such solutions must lie in the same orbit of a symmetry group preserving sudoku solutions.

We find necessary and sufficient conditions for completing an arbitrary 2 by n latin rectangle to an n by n symmetric latin square, for completing an arbitrary 2 by n latin rectangle to an n by n unipotent symmetric latin square, and for completing an arbitrary 1 by n latin rectangle to an n by n idempotent symmetric latin square. Equivalently, we prove necessary and sufficient conditions for the existence of an (n−1)-edge colouring of Kn (n even), and for n-edge colouring of Kn (n odd) in which the colours assigned to the edges incident with two vertices are specified in advance.