  A set X of partial words over a finite alphabet A is called unavoidable if
every two-sided infinite word over A has a factor compatible with an element of
X. Unlike the case of a set of words without holes, the problem of deciding
whether or not a given finite set of n partial words over a k-letter alphabet
is avoidable is NP-hard, even when we restrict to a set of partial words of
uniform length. So classifying such sets, with parameters k and n, as avoidable
or unavoidable becomes an interesting problem. In this paper, we work towards
this classification problem by investigating the maximum number of holes we can
fill in unavoidable sets of partial words of uniform length over an alphabet of
any fixed size, while maintaining the unavoidability property.
