Unavoidable arrays

An n × n array is avoidable if for each set of n symbols there is a Latin square on these symbols which differs from the array in every cell. We characterise all unavoidable square arrays with at most 2 symbols, and all unavoidable arrays of order at most 4. We also identify a number of general families of unavoidable arrays, which we conjecture to be a complete account of unavoidable arrays. Next, we investigate arrays with multiple entries in each cell, and identify a number of families of unavoidable multiple entry arrays. We also discuss fractional Latin squares, and their connections to unavoidable arrays. We note that when rephrasing our results as edge list-colourings of complete bipartite graphs, we have a situation where the lists of available colours are shorter than the length guaranteed by Galvin’s Theorem to allow proper colourings.