'Don't cry': Serena Williams consoles Dayana Yastremska; Simona Halep next

Serena Williams

Yastremska had never won so much as a single match at any Grand Slam tournament until this one, but she showed that she's capable of top-notch play during wins over 2011 U.S. Open champion Sam Stosur in the first round, then 23rd-seeded Carla Suarez Navarro in the second.

However, Halep - whose 2018 season was cut short thanks to a back injury - looked to be on better form on Saturday as she broke in the fifth game and again in the seventh to win the first set 6-2.

Other women's fourth-rounders set up for Monday: Naomi Osaka, the woman who beat Serena in last year's chaotic U.S. Open final, against No. 13 Anastasija Sevastova; 2017 U.S. Open runner-up Madison Keys against No. 6 Elina Svitolina; and two-time major champion Garbine Muguruza against 2016 U.S. Open runner-up Karolina Pliskova, who beat No. 27 seed Camila Giorgi 6-4, 3-6, 6-2 on Saturday night. "You did incredible. Don't cry". "I don't feel like I'm there yet", she said.

After comforting the world No.57 on court, Williams took her aside in the changing rooms to exchange some more words of encouragement.

Williams said later in her post-match press conference seeing her opponent in distress "broke my heart" and she recalled being in Yastremska's position when losing to Venus in the 2000 Wimbledon semifinals. I have to move her and give my best.

As an eight-year-old she had cheered Williams on to an Australian Open title a decade ago from her lounge room in Odessa, even simulating the American's strokes with her own racquet as she watched on television. Halep topped Venus Williams, 6-2, 6-3, also on Saturday.

Despite a reputation that precedes her when she walks onto center court, Williams is now ranked No. 16 in the world.

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Unlike any of Serena's foes until now, both Venus (with seven) and Halep (with one) have won a major title.

When asked if she has ever experienced a similar reaction when playing a match in her teens, Williams said she did as a young player and as an adult. "I think she's a good talent".

"It's always great to play against Serena because every time I have to learn something from the match. It shows she wasn't just there to play a good match, she was there to win", the American said. "If she does, I'll be there". By the end of the first set, the teenager had 13 unforced errors, nine more than Williams.

Things are set to get a whole lot tougher for Serena.

Were she to go on and win the tournament, the 37-year-old Williams would equal legendary Australian Margaret Court's all-time record of 24 women's grand slam titles.

What separates Williams from other top players?

When Yastremska loosened up and became more comfortable with the setting, though, there were signs of what has made her a much talked about rising star. 'I should know about this, but I don't know about this to comment on it.

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