On the intricacy of avoiding multiple-entry arrays

Let AA be any n×nn×n array on the symbols [n]={1,…,n}[n]={1,…,n}, with at most mm symbols in each cell. An n×nn×n Latin square LL on the symbols [n][n] is said to avoid  AA if no entry in LL is present in the corresponding cell of AA, and AA is said to be avoidable   if such a Latin square LL exists. The intricacy   of this problem is defined to be the minimum number of arrays into which AA 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.