MELBOURNE, Australia -- Serena Williams of the United States stayed on course to go for a record-breaking 24th Grand Slam title with a comprehensive 6-2, 6-1, third-round victory over 18-year-old Ukrainian Dayana Yastremska at the 2019 Australian Open on Saturday.

"I haven't played a lot since [the US Open in] New York," Williams said, during her post-match press conference. "But it is what it is. I'm here. I've trained in the off-season. You know, at this point I'm here. I'm here to play tennis, do what I do best. I play tennis best, so that's what I'm here to do."

Williams, seeded 16th this year, is making her 18th appearance at the Australian Open, and her seven titles at the major Down Under is an Open Era record. The American eased her way into the second week in Melbourne after a 67-minute victory over Yastremska in the first meeting between the two players.

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Yastremska is one of the fastest-rising youngsters on the WTA, as she picked up her first singles title in Hong Kong last autumn and will make her Top 50 debut after the Australian Open.

But Williams had the entire match firmly in her grasp, crushing 20 winners, including eight aces, and winning 85 percent of points on her first serve. Yastremska was unable to reach break point in the match, and though she had 13 winners of her own, the teenager was undone by 26 unforced errors and a 35 percent success rate on her second serve.

Williams will have a fascinating fourth-round encounter against either World No.1 Simona Halep or her sister, seven-time Grand Slam champion Venus Williams. Halep and Venus Williams will face off in a hotly anticipated third-round encounter on Saturday night. "Either way it will be a really intense match," said Serena Williams. "I'm looking forward to it."

"I honestly would love to face the World No.1," the 23-time major winner added. "At the same time, I would love for Venus to win. I think either way, regardless, I'll be ready for either opponent. Yeah, it will be great. I haven't played the World No.1 since I've been back, I don't think. So, yeah, it will be good."

Yastremska found herself in trouble from the get-go, dropping serve in the opening game after two double faults and a long forehand error on break point. Williams took the initiative and ran with it, per her champion’s mentality, racing to a 4-0 lead.

The Ukrainian teen found some good range on her forehand side in the latter half of the opener, and held in her next two service games. But the vaunted Williams service was impenetrable, as the American legend claimed 16 of her 20 service points en route to a comfortable one-set lead.

Yastremska got off to a good start in the second set, easing to an opening hold, closing out the game with an ace. But the Ukrainian was unable to extricate herself from a tricky spot in her following service game, and double faulted on Williams’s second break point to hand over a 2-1 lead to the long-time former World No.1.

From that juncture forward, Williams began to roll, using exceptional forehands to hold for 3-1, then parlaying her fantastic service returns to clinch another break for a 4-1 lead, after Yastremska could only dodge two of the three break points in that game.

Serving at 5-1, Yastremska again fended off one break point, which doubled as a match point, but a searing return winner by Williams gave the American a second chance to finish the job. There, a wide forehand mistake by Yastremska gave Williams the win, and the legend and the rising youngster shared a heartwarming hug at the net.

"As she was walking towards the net, I could tell [Yastremska] was quite upset," said Williams. "I kind of liked that. It shows she wasn't just there to play a good match, she was there to win. She wanted to win. That really broke my heart. I think she's a good talent. It's good to see that attitude."