From: rupert4050@my-deja.com
Subject: Re: Mathematics trivium
Date: Wed, 15 Dec 1999 15:25:24 GMT
Newsgroups: sci.math
Keywords: Graham's number

In article <838305$sqn$1@newsflash.concordia.ca>,
  mckay@cs.concordia.ca (MCKAY john) wrote:
> What is the exact size of the largest finite set studied in mathematics?
>
> JM
>
> --
> But leave the wise to wrangle, and with me
> the quarrel of the universe let be;
> and, in some corner of the hubbub couched,
> make game of that which makes as much of thee.
>

Graham's number is said to be the largest natural number to have
occurred in a mathematical proof. It is defined as follows.

Define m^n in the usual way.
Define m^^n to be m^(m^(... [m occurs n times] ... ^m)...)
Define m^^^n to be m^^(m^^(... [m occurs n times] ... ^^m)...)
etc.

Now I'm not sure if I get this next bit exactly right but it's
something like this:
Define f(0)=3
Define f(n+1)=3^...[f(n) ^'s] 3

Graham's number is f(64).

It's pretty big.


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From: David Bernier <bernier@my-deja.com>
Subject: Re: WORLD'S LARGEST NUMBER
Date: Mon, 27 Dec 1999 03:05:04 GMT
Newsgroups: sci.math

In article <3864FA09.78440F98@nospamaraknis.com>,
  Zubin Chandran <corezapr@nospamaraknis.com> wrote:
[...]
> Last time I checked the Guinness Book of World Records, the largest
> number EVER USED IN A PROOF i.e. having any marginally useful function
> was Graham's number. Graham's number is much bigger than Skewe's, and
> has to be expressed in arrow notation, as a recursive construction. The
> construction used grows much faster than the one you present.
>
> For those who haven't actually encountered it, there's a lot of material
> on the web relating to it. It's big enough that it boggles the
> imagination, there is no way to grasp it. I have been looking for the
> proof that it was used in, believe it was pertaining to Ramsey Theory,
> but was never able to find it. If someone can point me in the right
> direction, I would appreciate it.
[...]

Eric Weisstein's "Treasure Troves" site has a definition of
Graham's number in arrow notation.  Suppose we let
H_n := {0,1}^n.  We can think of H_n as the set of 2^n vertices of
a hypercube in n-dimensional space.  From H_n, we obtain
G_n, a graph whose vertices are the points of H_n and with
edges joining "pairs of corners" [it's not clear to me
what constitutes a pair of corners; maybe it's two vertices
separated by a Euclidean distance of sqrt(2)???].  In any case,
as I understand it, Graham's number is a "generalized Ramsey
number" upper bound, in the sense that any 2-coloring of
G_N forces a monochromatic complete graph on 4 vertices, where
N is Graham's number (upper bound).  From what I read, N is
the smallest known upper bound, though there may have been some
progress since the ('70's ??) paper was published.  Someone
acquainted with Ramsey theory or who has easy access to Math. Rev.
on-line might be able to elucidate further.

David Bernier
--
links of mathematical interest:
http://www.deja.com/~mathworld/


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From: oscwr@emory.edu (Cal Rice)
Subject: Re: largest meaningful number
Date: 14 Apr 1999 16:45:37 -0400
Newsgroups: sci.math
Keywords: Grapham's number

: SeraphSama (seraphsama@aol.comSPAMSUX) wrote:
: : The largest number used meaningfully is Graham's number, which is much, much
: : bigger than a googolplex, and 10^200.
 
   Never mind defining Graham's number here.  A definition can be found
   at 
 
        http://public.logica.com/~stepneys/cyc/g/graham.htm
 
[deletia --djr]