The Hungarian mathematician PaulErdos wrote a lot of joint papers. One's Erdos Number is defined as the length of the shortest path between you and Erdos via peer-reviewed papers. Hence Erdős Erdős number was 0, if you are one of his co-authors then your Erdős number is 1. If you aren't, but you co-wrote a paper with one of them, then your Erdős number is 2. And so on.  If there is no such path then the convention is to say that your ErdosNumber is infinity, whether you give a flying fsck about mathematics or not.

Lamentably, it is no longer possible for the number of people with Erdős number 1 to increase.

The canonical reference for this concept is the ErdosNumberProject, where JerryGrossman offers long lists of people with an Erdős number of up to 2.  If you've published a peer-reviewed paper, your number may be lower than you think.

Some people, with a sense of fun, play the same game using any publicly accessible joint authorship.  This rapidly becomes pretty pointless, especially taking wiki pages into consideration. Others suggest that it is disrespectful of Erdős to pretend that there is meaning in a non-mathematical version of this, and suggest using a different "root node", such as Ward (WardNumber) or DonKnuth.

Film fans have a similar concept relating to KevinBacon. Erdős numbers were there first (and were the inspiration for Bacon numbers).

Erdős himself has a Bacon number of 4:
* http://oracleofbacon.org/cgi-bin/oracle/movielinks?firstname=Kevin+Bacon&game=1&secondname=Paul+Erdos.
It is unlikely that Kevin Bacon has an Erdős number.

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Since college, I've only known one 'real' Mathematician socially. I checked the list today and he has a ErdosNumber of 2. Either it's a SmallWorld or I just got lucky.

''So whom do you know?''
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Here is a map of Erdős' 500-something collaborators...
http://www.orgnet.com/Erdos.html
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'''Decreased ErdosNumber for sale!''' http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&item=3189039958

BillTozier has an ErdosNumber of 4 and is offering to collaborate (for a fee) on a paper with anyone seeking an ErdosNumber of 5.  

''Given the large number of folks with ErdosNumber''''''s of 4 or less, this "service" seems to not be worth much.''

I think it's pretty funny. But no, to get an EN of 5 isn't worth much, which is why it hasn't been bid sky high. It would be headline material if it were for an EN of 2, and the bidding would, I predict, be fierce.

''Heck, I just did some poking around, and discovered I have an ErdosNumber of 4.  Details to be posted soon on my HomePage. -- ScottJohnson''
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There's more going on in BillTozier's auction than meets the eye. So says him, at least. He's posted an addendum to the text, and is blogging the auction dynamics. -- ''Bill'' (in ThirdPerson)
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I know someone with an ErdosNumber of 2, who was bemused when I told him about the auction. If anyone wants to make an offer.... :-)

''I have an ErdosNumber of 2, and I'm slightly saddened by the episode. Still, if the resulting work is legitimate then I have no grounds for complaint.''