Unary Pattern Avoidance in Partial Words Dense with Holes

A partial word is a sequence of symbols over a finite alphabet that may have some undefined positions, called holes, that match every letter of the alphabet. Previous work completed the classification of all unary patterns with respect to partial word avoidability, as well as the classification of all binary patterns with respect to non-trivial partial word avoidability. In this paper, we pose the problem of avoiding patterns in partial words very dense with holes. We define the concept of hole sparsity, a measure of the frequency of holes in a partial word, and determine the minimum hole sparsity for all unary patterns in the context of trivial and non-trivial avoidability.