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 → ∞.