Ramsey Theory and Distortion
Ted Odell
February 6, 2008
The pigeonhole principle is the simplest Ramsey Theorem. The rigorous
statement of the the principle is that if X and Y are finite and |X| is less than
|Y |. There is no injection from Y to X.
A simple application is that if 150 people shake hands with one another then
at least two people must have shaken the same hand. This is true because any
person could have shaken between 0 and 149 hands. So at a first glance you
might think that each person might have shaken a distinct number of hands.
But if someone shook 0 hands then no one could have shaken 149. Therefore
by the pigeonhole principle two people must have shaken the number of hands.
Imagine n people in a room, some of them shake hands and some of them
don’t. Consider all the possible pairs of people. Color a pair pink if they did not
shake hands and color green if they did shake hands. Another way of visualizing
this problem is as a graph. Start with n nodes (one for each person) and add a
pink edge between a pair of people if they didn’t shake hands and a green edge
if they. Define the number R(n) to be the smallest number of vertices such that
you can find a complete graph of size n that is either all pink or all green.
We know that r(2) = 2, r(3) = 6, and r(4) = 18. We know r(5) is between
43 and 49. You might think you could use a computer to search through all the
possible graphs of order 43 to determine R(5).
29 02 graphs of size 43 28 2 atoms in the universe
Ramsey showed that R(n) always exists for finite n. Further more he showed
that you can even extend this to find infinite complete subsets. Prove this for
notes (use sequences)
Ramsey theory is about taking heterogeneous structures and finding homogeneous substructures.
p
Euclidean geometry Rn has the distance function (x, y) 7→ (x1 − y1 )2 + . . . + (xn − yn )2 .
There are other geometries such as the Taxicab geometry with the Manhattan
distance function (x, y) 7→ |x1 − y1 | + . . . + |xn − yn |. Another geometry has the
the distance function (x, y) 7→ max(|x1 − y1 |, . . . , |xn − yn |). Any reasonable
distance function satisfies the triangle inequality so the unit sphere is always
convex. Knowing this you can create a geometry by constructing a bounded
convex set that contains the origin and use that to define the distance function.
How does this connect to Ramsey Theory? Given a strange geometry we
want to find a subspace that looks locally like Euclidean Space. explain 2
coloring sphere, infinite dimensional ribbon covering, and Dr Odell’s research.
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There are other Ramsey Theorems. If you split the natural numbers into
two colors then one of the colors must contain arbitrarily long arithmetic progressions. In 2004 Ben Green and Terry Tao showed if you two color the natural
numbers one of them must contain arbitrarily long progressions of primes.
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