This theorem actually has a name, it was
called the Happy Ending theorem. And what
happened was that a woman mathematician,
Esther Klein, picked up the
problem when she was abroad and brought
it back and Erdős and Szekeres
thought about it and it turns out that
shortly thereafter
Szekeres and Esther Klein got married,
and they were married until- in their 90s.
So if I put five points on the table
here in the plane so that no three are
in a straight line; so that's one way to
do it, or I could do it like this, or I
could do it a different way. What you're
looking for, you're looking for a
convex set of four points. Now what does
convex mean? Convex means, as you're moving
around the polygon, say you're going from
here to here to here so I went all the
way around the edges, you're always
turning left, you never turn right. This
is the idea of a convex polygon. And a
polygon that's not convex is something
like this: so if I were to go around the
outside of this polygon, here I turn
right, then I turn left, then I turned left,
then I turned left, so it kind of dented
in. So you're looking for a convex
polygon; now here's a set of five points,
could I have a convex? Yes, here's a
convex one right here. How about this one?
Well that's a little harder but here's one.
(Brady: You don't have to use all five points?)
No, you only get four points. So the question
is now, if I just gave you four points
like this then there is no convex 4-gon
here because you got to use all
four points and so you'd be stuck. But
five points you have to get a convex
4-gon and there's a very beautiful
nice proof of that. If the outside only
had four or five points then you could
use those points and get your convex
four-sided figure, but if it didn't; so if
you had a triangle and two points on the
inside okay? So look at these five points.
Then if you just take this line and
extend it through then it will hit a
side over here and then these four
points form a convex n-gon. So the
question is, suppose you wanted not a
convex 4-gon, you wanted a convex
five sided figure. If you put enough
points down must you get a five? Yes.
And the answer is yes. If you put nine
points down,
no three in a straight line, you must
form a convex five sided figure
someplace. And nine is the best possible,
with eight you can prevent it, with eight you
can prevent it. And just very recently
people showed that if you put 17 points
down you must get a convict six-sided
figure, and 16 is not enough. So it's a
beautiful conjecture of Erdős and
George Szekeres, who was his longtime
collaborator, that if you want a convex
n sided figure, an n-gon, then it's
enough to put 2 to the n minus 2 plus 1
points. That should imply that you would
get a convex n-gon. And that would be
best possible. There is a way to put only
2 to the n minus 2 and not get one. So if
n is equal to say 6, you want a convex
6-gon, then 2 to the 6 minus 2 plus 1
which is 17, and that was that recent
result - 17's enough. And if you put n is
equal to 4, then you want a convex
4-gon, 2 to the 2 plus 1 is 5. That's our 5
okay? So it's- the guess is that that's
exactly the right answer, and I for
$1,000, kind of in Erdős' name, to
prove that that's the right answer or
prove it's not the right answer.
(1000 bucks? Of your money?) - Prove or
disproved. Well I have some, yes, I have
some Erdős money left for this
emergency in case people solve his
problems, right? - (So that's not- that's not)
(that's not fact yet? That's just a-) - No,
this is a conjecture.
There are many papers and surveys and
and it's just one of these kind of
innocent geometrical problems; again in
the spirit of Ramsey theory which shows
that in a complex situation there's a
certain order within it. In the complex
situation, there's a bunch of endpoints in the
plane and the order is a n-gon
that's convex. It was known to n equals 4
was right away- right away. And 5 was done
30-40 years ago. And 6 was just done,
and they used a fair amount of computer
calculation because you have to look at
all the different configurations of
putting 17 points down in the plane and
there are a lot of different
possibilities and you want to check that
each one of them must have a convex
6-gon, and they did and that's done. But
now the next case is going to be on
computation so you need
your brain once again. Right now
computers aren't as- well, aren't up to it
for replacing our brains, at least not up to it yet.
...that stack is 7.6 trillion
3s high. And you start
from the top and work your way down and
you get an almighty number, you get a
number that is absolutely off the chart.