The theorem of Fraenkel and Simpson states that the maximum numberof distinct squares that a word w of length n can contain isless than 2n. This is based on the fact that no more than twosquares can have their last occurrences starting at the sameposition. In this paper we show that the maximum number of the lastoccurrences of squares per position in a partial word containing onehole is 2k, where k is the size of the alphabet. Moreover, weprove that the number of distinct squares in a partial word with onehole and of length n is less than 4n, regardless of the size ofthe alphabet. For binary partial words, this upper bound can bereduced to 3n.
