Latin cubes of even order with forbidden entries

We consider the problem of constructing Latin cubes subject to the condition that some symbols may not appear in certain cells. We prove that there is a constant γ>0 such that if n is even and A is a 3-dimensional n×n×n array where every cell contains at most γn symbols, and every symbol occurs at most γn times in every line of A, then A is avoidable; that is, there is a Latin cube L of order n such that for every i,j,k∈{1,…,n}, the symbol in...