Before we leave you, here's where we're up to in the other matches around the grounds in New York.

On Louis Armstrong Stadium, Frances Tiafoe has just dominated the first set tie-break to land the first blow against Daniil Medvedev in what is shaping up to be an extremely tight contest.

That's been a real tussle so far. Former champion Medvedev trailed by a break and 4-2 but turned things around to give himself the chance to serve it out - but Tiafoe won seven of the eight tie-break points.

On Grandstand, another home hope, Alex Michelsen, is in control against Argentine Tomas Martin Etcheverry, leading by a set and a break now.

Elsewhere, Britons Harriet Dart and Maia Lumsden have lost their second-round match in the women's doubles to Cristina Bucsa and Nicole Melichar-Martinez 6-2 6-2.

Image source, Getty Images

If you weren't with us earlier, women's world number one and defending US Open champion Aryna Sabalenka was a straight-set winner against American Taylor Townsend on Arthur Ashe Stadium.

Sabalenka took a first set which contained seven breaks of serve and recovered from a break down in the second to win 6-4 6-3 in one hour and 15 minutes.

Carlos Alcaraz on his performance against Tommy Paul: "I played my game, I tried to play my style. Tommy is such a talented player, his serves are pretty dangerous.

"Sometimes it's pretty difficult to control the ball when it comes to you.

"I tried to be ready for the battle, to run if it matters, to be aggressive all the time. I think I did really well, I just put so much pressure on the return game which for me helped a lot.

"You play relaxed, you play calm, so I'm just really happy with the performance today."

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Jeff Tarango
Former American tennis player on BBC Radio 5 Live

How different this draw would have been if Jannik Sinner had entered, Carlos Alcaraz would have been third seed, instead of the second. It would have definitely changed everything.

Image source, Reuters

It's shaping up to be a real scrap over on Louis Armstrong Stadium.

American Frances Tiafoe goes from a break down to a break up - but he can't consolidate his position and Daniil Medvedev restores parity at four games apiece.

Over on Grandstand, Alex Michelsen has stolen the first-set tie-break from Argentine Tomas Martin Etcheverry, recovering from two set points down to eventually take it 8-6.

That's gone down very well with the home supporters!

Jeff Tarango
Former American tennis player on BBC Radio 5 Live

There you see his electric smile, he's really happy to be there and be back.

What's more amazing is match point, second serve, that's the first time he's hit a second serve over 110mph and he does so right then. It was a complete surprise and obviously a great champion.

Just stunning!

Carlos Alcaraz with a 96mph forehand winner, cross-court and inch-perfect to bring up match point.

Tommy Paul is not giving up yet and he brings up two break points.

He wastes the first by putting a backhand slice into the net before Carlos Alcaraz comes alive with laser of a forehand down the line and then advances to the net to finish the point.

Deuce.

This wasn't in the script, Tommy Paul is up 0-30 on the Carlos Alcaraz serve...

Tommy Paul does what he has to do.

He holds to love for the first time in the match.

The players will have a quick sit down and a drink, after which, Carlos Alcaraz will serve for the match.

One game away.

Carlos Alcaraz crushes a cross-court forehand winner to move to the brink of the last eight at Flushing Meadows.

Tommy Paul will try to make him wait but, at this point, it is hard to see how it would be anything more than a slight delay.

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Not quite as simple as it might've been after racing into a 40-0 lead but Tommy Paul holds.

Carlos Alcaraz entertains the crowd with a through-the-legs shot but Paul had gone long anyway... I'm not 100% sure the Spaniard knew it was long, though.

Anyway, Alcaraz is still two games away from the quarter-finals.

Jeff Tarango
Former American tennis player on BBC Radio 5 Live

Amazing get there by Tommy Paul. Alcaraz had that overhead, crushed it. But then he had a relatively easy backhand pass and missed it. I mean that was the easiest shot he had the whole rally.

Big cheers from the crowd on Arthur Ashe as Tommy Paul returns Carlos Alcaraz's overhead with interest.

Good as it was, Alcaraz still manages to win the point. You really do have to feel for his opponents when he's in this kind of mood.

Paul doesn't even attempt to get to a superbly disguised drop shot next point and Alcaraz holds to 15.

Image source, Reuters

The other two ongoing men's fourth-round matches are on serve in the first set following an exchange of early breaks.

Home favourite Frances Tiafoe dropped serve in his opening game but produced an immediate response against Russian seventh seed Daniil Medvedev, who triumphed here in 2021.

Medvedev has just saved three more break points, thanks to some excellent serving, to level at 2-2.

Another American, Alex Michelsen, led by a break and 3-0 against Argentine Tomas Martin Etcheverry - but they are currently tied at five games apiece.

Huge cheers from the home crowd as Tommy Paul secures the lesser spotted comfortable hold with an ace out wide.

The 20th seed is still, just about, hanging in there.

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I was just thinking it had been a little while since we'd seen a really eye-catching winner from Carlos Alcaraz - well, a couple of games anyway.

He must've been thinking the same as he smokes a forehand beyond a helpless Tommy Paul.

Break consolidated.

Jeff Tarango
Former American tennis player on BBC Radio 5 Live

He just made three unforced errors at break point, at deuce point there is marvellous. Tommy Paul threw everything at him, and the kitchen sink. And Alcaraz came up with something better on every shot.

Tommy Paul can't do a whole lot more...

He's into the net quickly and hammers the ball at Carlos Alcaraz, who produces another ridiculous reaction volley.

That's the break for Alcaraz. Paul can't quite believe he's lost that point.