# Latin cubes of even order with forbidden entries

## TL;DR

This paper proves that for even-order Latin cubes with limited forbidden entries per cell and line, it is always possible to construct a Latin cube avoiding these forbidden symbols, extending combinatorial understanding of Latin structures.

## Contribution

It establishes the existence of Latin cubes avoiding specified forbidden entries under certain density constraints for even orders, a new result in combinatorial design theory.

## Key findings

- Existence of avoiding Latin cubes for even orders with bounded forbidden entries.
- A constant gamma ensures the avoidance is always possible under the given conditions.
- The result applies to 3-dimensional arrays with controlled symbol frequency.

## 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 $n=2t$ 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 {\em 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 $A$.