• Former champion drops her first set of US Open • Defending champion Sloane Stephens also through

As one favourite after another has staggered out of this US Open, Serena Williams has looked resolute and indomitable in pursuit of a seventh title – until she encountered someone who hits the ball as hard as she does, Kaia Kanepi, who almost derailed her campaign on Sunday.

Williams had lost only 11 games in three matches over three hours and 28 minutes coming into the fourth round. She left, after another hour and 36 minutes, a little shaken but relieved to have posted a 6-0, 4-6, 6-3 win. Her serve got her out of a lot of trouble, 18 aces draining the unseeded Estonian’s spirit.

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“It wasn’t an easy match at all,” she said courtside. “She’s had a lot of good wins in her career. I’m just happy to get through it, to be honest.”

The American’s 18-minute blitz in the first set equalled the quickest of the tournament, that by Lara Arruabarrena over Kateryna Kozlova on the first day, but she then encountered the most stubborn fightback, as Kanepi, her left shoulder and right elbow strapped, shifted the average speed of her returns up from 50 miles an hour when being bagelled to 70mph in the second.

The aces began to flow in the third and Kanepi could not rediscover her earlier zest.

While Williams has been away, recovering from the difficult arrival of her first child a year ago, others have jostled unconvincingly for dominance. Caroline Wozniacki won the Australian title that Williams was unable to defend, Simona Halep broke through at Roland Garros, and at Wimbledon Angelique Kerber lifted the title by beating the American in her second comeback major. Those three title-holders have been knocked out here in the first week, the world No1, Halep, falling to Kanepi, who had not dropped a set before Sunday.

The opposition will get stronger in the second week, with the eighth seed, Karolina Pliskova – the last player to beat her here – waiting for Williams in the quarters, and the possibility of a meeting with the defending champion, Sloane Stephens, in the semi-finals. Stephens looked comfortable as she beat Elise Mehrtens 6-3, 6-3 in their fourth-round match on Sunday night.

Expectations that the men’s quarter-finals would bring together last year’s US Open finalists were shredded when Rafael Nadal won after a dip and a struggle and Kevin Anderson was a brave but exhausted loser. However, there will be a re-run of the French Open final, as Nadal’s opponent tomorrow is Dominic Thiem.

Nikoloz Basilashvili, the last unseeded player on this side of the draw, took only one game off Nadal in their only previous meeting, at the French Open last year, but he played probably the match of his life on Arthur Ashe Court on Sunday, and Nadal had to soak up stern if flickering resistance to win 6-3, 6-3, 6-7 (6), 6-4.

“Two tough matches in a row” was Nadal’s answer when asked if this was harder than his long four-setter against the Russian Karen Khachanov on Friday. “I had big chances at the beginning but he played fantastic. A lot of credit to him, hitting the ball very, very strong. I think I improved my level at the end. I did what I had to do. I found a way.” As for Thiem, he said, “He is a great opponent, big serve. It will be a tough one.”

The Austrian, a first-time finalist at Roland Garros this year, has been strangely out of sorts over the first week of the year’s final major and the ninth seed concedes a change of surface will probably do him no favours against the defending champion.

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“On clay it’s one of the biggest challenges in sports to beat this guy or to compete with him,” Thiem said. “I hope that it’s a little bit more comfortable on hard court but I’m not sure.”

Their head-to-head favours Nadal 7-3, the most recent meeting that final in Paris, when the undisputed master of clay allowed Thiem only nine games in two hours and 42 minutes. In fact, they have played all their 10 matches on clay but Nadal has a pedigree across all the surfaces.

It was difficult not to feel sorry for the wholehearted Anderson when Thiem put him to the sword for only the second time in their eight meetings, 7-5, 6-2, 7-6 (2) in two hours and 37 minutes.

The 6ft 8in South African, who went three sets with Nadal in last year’s final and the same against Novak Djokovic in the final at Wimbledon this year, was not bothered that he was again ignored for a place on the tournament’s show court.

“I understand the guys out on Ashe are superstars of our sport,” he said. “They get a lot of people watching. I need to win a few more tournaments to get out there. I wanted to be here right till the end and put myself in contention of winning my first major. It wasn’t meant to be.”