Latin Cubes with Forbidden Entries

TL;DR

This paper proves the existence of Latin cubes avoiding certain forbidden entries under specific density conditions, extending combinatorial constructions for Latin structures with restrictions.

Contribution

It establishes a new result showing that Latin cubes can be constructed to avoid forbidden entries when the forbidden pattern is sufficiently sparse.

Findings

Existence of Latin cubes avoiding forbidden entries under density constraints

Construction method for Latin cubes with restricted entries

Bound on the density of forbidden symbols for avoidability

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 $n=2^k$ and $A$ is $3$-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 $A$.