On avoiding some families of arrays

An n × n array A with entries from {1, . . . , n} is avoidable if there is an n × n Latin square L such that no cell in L contains a symbol that occurs in the corresponding cell in A. We show that the problem of determining whether an array that contains at most two entries per cell is avoidable is NP-complete, even in the case when the array has entries from only two distinct symbols. Assuming that P = NP, this disproves a conjecture by Öhman. Furthermore, we present several new families of avoidable arrays. In particular, every single entry array (arrays where each cell contains at most one symbol) of order n ≥ 2k with entries from at most k distinct symbols and where each symbol occurs in at most n − 2 cells is avoidable, and every single entry array of order n, where each of the symbols 1, . . . , n occurs in at at most n 6 cells, is avoidable. Additionally, if k ≥ 2, then every single entry array of order at least n ≥ 4, where at most k rows contain non-empty cells and where each symbol occurs in at most n− k + 1 cells, is avoidable.