MELBOURNE   -  As Serena Williams made her way out to the Australian Open’s main court during pre-match introductions, the voice booming over the stadium’s speakers implored spectators to welcome “the world’s No. 1 player — from Romania, Simona Halep.”

Williams made way for her fourth-round opponent Monday night. A couple of hours later, when Williams wrapped up her 6-1, 4-6, 6-4 victory over Halep in a contest filled with momentum swings and fantastic play by both, it was clear which woman really is No 1, regardless of what the current rankings say. “It was a really intense match, and it was some incredible points,” said Williams, who improved to 9-1 against Halep and now meets 2016 U.S. Open runner-up Karolina Pliskova in the quarterfinals. “I love playing tennis and I love to play here and I love to be back out here.”

Halep offered up the first real test Williams has faced at this year’s Australian Open, where she is bidding for an eighth trophy, pushing her throughout a back-and-forth matchup that was a fascinating mix of power, court coverage and shotmaking. It wasn’t until Williams saved three break points in a monumental game to hold for 3-all in the third set that she gained control.

Pliskova advanced earlier Monday with a 6-3, 6-1 victory over two-time major champion Garbine Muguruza. The other quarterfinal on that side of the women’s bracket features No. 4 Naomi Osaka against No. 6 Elina Svitolina. Osaka, who beat Williams in last year’s U.S. Open final, edged No. 13 Anastasija Sevastova 4-6, 6-3, 6-4 in the fourth round, while Svitolina got past 2017 U.S. Open finalist Madison Keys 6-2, 1-6, 6-1.

Novak Djokovic is confident he will be ready to face Japan’s Kei Nishikori in the Australian Open quarter-finals after a gruelling last-16 battle against Russia’s Daniil Medvedev. The 31-year-old Serb earned a 6-4 6-7 (5-7) 6-2 6-3 victory over the 15th seed in Melbourne. Djokovic was not at his fluent best, but outlasted Medvedev, 22, to win in three hours and 15 minutes. “A couple of things have surfaced after a match like this,” he said. “I’ll see how the body reacts.”

Medvedev also struggled physically and grew frustrated as he paid the price for only taking two of nine break points. However, his resistance continued until the end, saving two match points before Djokovic took the third with a wonderful cross-court winner. Six-time champion Djokovic will play Nishikori in the last eight after the eighth seed beat Spain’s Pablo Carreno Busta in controversial circumstances in a five-hour match decided by a super tie-break.

Top seed Djokovic, who tumbled over in the third set and fell again in the fourth, looked uncomfortable in the final stages of the match. “I didn’t feel so great in the final 20 minutes or so,” he said. “A bit of fatigue, a bit of hurt in the back, but nothing major. I’m confident I can be ready.”

In other men’s matches, 2016 Wimbledon runner-up Milos Raonic knocked off No 4-seeded Alexander Zverev 6-1, 6-1, 7-6 (5), and next takes on No. 28 Lucas Pouille, a 6-7 (4), 6-4, 7-5, 7-6 (2) winner over No. 11 Borna Coric. No. 23 Pablo Carreno Busta left the court yelling at the chair umpire after a disagreement in the concluding tiebreaker of his 6-7 (8), 4-6, 7-6 (4), 6-4, 7-6 (8) loss to No. 8-seeded Kei Nishikori in a 5-hour, 5-minute marathon that was the longest match of the tournament.