Latin Cubes with Forbidden Entries
Carl Johan Casselgren, Klas Markstr\"om, Lan Anh Pham

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 such that if and is -dimensional array where every cell contains at most symbols, and every symbol occurs at most times in every line of , then is {\em avoidable}; that is, there is a Latin cube of order such that for every , the symbol in position of does not appear in the corresponding cell of .
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