Timothy Y. Chow
tchow at alum.mit.edu
Wed Mar 29 10:05:32 EST 2006
Harvey Friedman <friedman at math.ohio-state.edu> wrote:
> Graham's number is upper bounded by the value of the 4th (I may be off
> by 1) function in the Ackermann hierarchy at a fairly small argument.
I don't think this is correct. I think Graham's number is something like
A_(A_(A_(...A_4(3)...)(3))(3))(3), with 64 levels of subscripting.
> Of course, Graham's number has some special mathematical significance of
> a different kind than my numbers. So Graham's number of course REMAINS
> quite important, despite its PUNY size.
Graham's number doesn't have any direct relation to proof-theoretic
issues, but in one sense it is similar to the numbers in your "long finite
sequences" paper, in that it is a *bound* for an innocuous-looking
combinatorial problem. In fact, it is an *upper* bound for a certain
Euclidean-type Ramsey number whose true value is conjectured by some
people to be 6. So its intrinsic significance is doubtful; it's really
just a measure of how inadequate our proof techniques are for such a
The problem in question, by the way, is to take a hypercube in R^n, join
every pair of vertices with an edge, and two-color the edges so as to
avoid a monochromatic *planar* K_4. Graham's number is bound on the
critical dimension n.
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