Latin Squares with Forbidden Entries

Abstract

An $n \times n$ array is avoidable if there exists a Latin square which differs from the array in every cell. The main aim of this paper is to present a generalization of a result of Chetwynd and Rhodes involving avoiding arrays with multiple entries in each cell. They proved a result regarding arrays with at most two entries in each cell, and we generalize their method to obtain a similar result for arrays with arbitrarily many entries per cell. In particular, we prove that if $m\in {\Bbb N}$, there exists an $N=N(m)$ such that if $F$ is an $N\times N$ array with at most $m$ entries in each cell, then $F$ is avoidable.