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If the minimum vertex degree of H is greater than \\binom{n-1}{2}-\\binom{2n/3}{2}, then H contains a perfect matching. This bound is tight and answers a question of Han, Person and Schacht. More generally, we show that H contains a matching of size d\\le n/3 if its minimum vertex degree is greater than \\binom{n-1}{2}-\\binom{n-d}{2}, which is also best possible. 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