An individual's Erdős–Bacon number is the sum of one's Erdős number—which measures the "collaborative distance" in authoring mathematical papers between that individual and Hungarian mathematician Paul Erdős—and one's Bacon number—which represents the number of links, through roles in films, by which the individual is separated from actor Kevin Bacon.
Daniel Kleitman, a mathematician at MIT, was an advisor for the movie Good Will Hunting and appeared briefly as an uncredited extra. Minnie Driver appeared in that movie, and she also appeared in Sleepers with Kevin Bacon. So Kleitman's Bacon number is 2. On the other hand, he coauthored a paper with Erdős. This gives him an Erdős–Bacon number of 3. This would probably be the minimum over all existing Erdős–Bacon numbers.
For a time, the sole person with the lowest known Erdős–Bacon number was popular string theorist Brian Greene. He appeared in Frequency with John Di Benedetto, who was in Sleepers with Kevin Bacon, for a Bacon number of 2. He also wrote a paper with Shing-Tung Yau, who wrote a paper with Ronald Graham, who wrote a paper with Paul Erdős, for an Erdős number of 3 and a combined Erdős–Bacon number of 5.
Brian Greene was later topped by Dave Bayer, mathematical consultant to A Beautiful Mind who received a minor role on screen in the movie. Rance Howard was also in A Beautiful Mind and in Apollo 13 with Kevin Bacon to give Bayer a Bacon number of 2. Bayer wrote a paper with Persi Diaconis, who has an Erdős number of 1 due to a jointly authored 1977 Stanford University technical report, later published in a 2004 compilation. So Bayer's Erdős–Bacon number is 4.
This is when it gets really weird
Update: Ben Rosenbaum was kind enough to correct me on and say that he didn't dream it up, he merely re-invented i it.Apparently, it was dreamed up by a SF writer Ben Rosenbaum, at WorldCon a few years ago. As mentioned on BoingBoing
Danica McKellar, most famous for her role as Winnie Cooper in The Wonder Years, has an Erdős–Bacon number of 6, having coauthored a mathematics paper published while an undergraduate at UCLA. This gives her the lowest known Erdős–Bacon number for a professional actor/actress. Her paper gives her an Erdős number of 4, and she and Bacon have both worked with Margaret Easley.
Natalie Portman has an Erdős–Bacon number of at most 9. She collaborated with David A. Boas (using her birth name, Natalie Hershlag), who has a collaboration path leading to Frank Harary, who has an Erdős number of 1. Bacon and Portman have both worked with Matt Dillon, giving Portman a Bacon number of 2 and an Erdős number of at most 7.