# Latin Cubes with Forbidden Entries

### Abstract

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 $\gamma > 0$ such that if $n=2^t$ and $A$ is a $3$-dimensional $n\times n\times n$ array where every cell contains at most $\gamma n$ symbols, and every symbol occurs at most $\gamma n$ 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 $1\leq i,j,k\leq n$, the symbol in position $(i,j,k)$ of $L$ does not appear in the corresponding cell of $A$.