Overlap-freeness in infinite partial words

We prove that there exist infinitely many infinite overlap-free binary partial words containing at least one hole. Moreover, we show that these words cannot contain more than one hole and the only hole must occur either in the first or in the second position. We define that a partial word is k-overlap-free if it does not contain a factor of the form xyxyx where the length of x is at least k. We prove that there exist infinitely many 2-overlap-free binary partial words containing an infinite number of holes.

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