And that’s what makes this week’s 39th edition of the Ryder Cup seem even bigger.

Europe has the No. 1 player in the world on its side — Rory McIlroy — for the first time in two decades. Jim Furyk referred to the 23-year-old from Northern Ireland as a ‘‘marked man’’ because of his staggering success dating to his record eight-shot win at the PGA Championship, his second major in as many years.

‘‘He’s the best player in the world right now, and he’s going to be the toughest guy to beat,’’ Furyk said.

The United States counters with Tiger Woods, whose game has been restored enough for him to win three times on the PGA Tour this year. Woods is even starting to figure out this Ryder Cup business, producing a winning record (6-3-0) the last two times he has played.

The quality of the rosters doesn’t stop with golf’s two biggest stars, however. For the first time in Ryder Cup history, the 24 players who will be at Medinah Country Club in the Chicago suburbs all are among the top 35 in the world ranking.

Already the most intense competition in golf, this Ryder Cup has all the trappings of a heavyweight title fight.

‘‘It does seem that way,’’ Luke Donald said. ‘‘McIlroy is on a phenomenal run. Really, everyone is coming into form.’’

The Americans have four players who will be making their Ryder Cup debut, though these ‘‘rookies’’ have some major experience handling pressure: Masters champion Bubba Watson, US Open champion Webb Simpson, and Keegan Bradley, who won the PGA Championship last year.

‘‘I’ve been looking at the US team for a few weeks now and just going, ‘Wow. These guys look really strong and really good,’ ’’ said Graeme McDowell, the Ryder Cup hero in Wales two years ago when he won the decisive point at Celtic Manor. ‘‘And there’s so much young blood on there now who are right up for these things.

“There’s a good buzz. It’s great. I think it’s set up to be an awesome Ryder Cup.’’

Europe has a team so strong that Padraig Harrington was left off for the first time since 1997, and the defending champions have only one rookie. That would be Nicolas Colsaerts of Belgium, one of the longest hitters in the game whose lone win this year came at the World Match Play Championship in Spain.

Along with McIlroy, Europe features the only other three players to be No. 1 in the world since Woods abandoned the top spot two years ago: Donald, Lee Westwood, and Martin Kaymer. Sergio Garcia, who has never lost a foursomes match in his five Ryder Cups, is back on the team after missing in 2010. One of the captain’s picks was Ian Poulter, who has won all three of his singles matches before they reached the 17th hole.

The chatter at most Ryder Cup matches is which team looks better on paper.

This time, they’re both mint.

‘‘The US team has a number of players high up in the ranking, as does Europe,’’ Phil Mickelson said. ‘‘We’re going to see some high-quality golf at the Ryder Cup.’’

Adding to the star power are the captains.

Jose Maria Olazabal is such a central figure in the Ryder Cup, dating to his ‘‘Spanish Armada’’ partnership with the late Seve Ballesteros, that when the European Tour committee set out to identify the next captain, it didn’t even bother meeting. He is the only European captain ever selected by a phone call.

The Americans have Davis Love III, a widely respected figure on the PGA Tour who made the cup-clinching putt in his Ryder Cup debut in 1993 and played on six teams.

The Americans would seem to have one advantage simply by playing before a home crowd. And for Woods, it will feel like a home course. Medinah has hosted five major championships, the last two won by Woods.

‘‘I personally like Medinah,’’ Woods said with a grin.

The Americans have lost only once at home in the last 15 years — in 2004 at Oakland Hills — and most Ryder Cup veterans believe playing at home is worth a point.

Medinah might look like a big stage, though it’s not likely to play like a major test.