[FOM] Some questions regarding irrational numbers
Timothy Y. Chow
tchow at alum.mit.edu
Sun Mar 1 21:51:25 EST 2009
Lasse Rempe wrote:
> Informally, my question really is: is there a number, conjectured but
> not known to be an integer, for which the first thought of a high-school
> student or beginning undergraduate would be "why don't we just compute
> it?"
Well, then, the Turan hypergraph example might work for you---the only
problem being that it might be hard to explain the example to a
high-school student or even an undegraduate. Let me say precisely what it
is. If it has the right flavor, one could hunt more diligently for a more
elementary example of this type.
I'm following Chung and Graham's presentation in their book "Erdos on
Graphs." By an "r-graph" they mean an r-uniform hypergraph, i.e., a
family of r-element subsets of an n-element set. Let us define t(n)
to be the largest integer t such that there exists a 3-graph on n vertices
with t edges which does not contain any complete 3-graph on 5 vertices.
Then Turan conjectured that
lim_{n->oo} (n choose 3)/t(n) = 4.
In principle this is not that hard to explain to an undergraduate, but I
would predict a lot of glazed-over eyes because it would be so far removed
from their mathematical experience that the definition would seem totally
artificial and unmotivated.
A more technical objection is that I'm not sure that anyone has gone so
far as to predict a rate of convergence, so maybe the limit (which can
easily be proved to exist) doesn't count as being "computable."
Tim
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