On the intricacy of avoiding multiple-entry arrays

Let A be any n×n array on the symbols [n]={1,…,n}, with at most m symbols in each cell. An n×n Latin square L on the symbols [n] is said to avoidA if no entry in L is present in the corresponding cell of A, and A is said to be avoidable if such a Latin square L exists. The intricacy of this problem is defined to be the minimum number of arrays into which A must be split in order to ensure that each part is avoidable. We present lower and upper bounds for the intricacy, and conjecture that the lower bound is in fact the correct answer.