1. Colorings of pairs. I
There are several possible ways in which one can try to generalize Ramsey’s theorem to larger cardinalities. We will discuss some of these generalizations in upcoming lectures. For now, let’s highlight some obstacles.
Theorem 1 (
-Kakutani)
In fact,
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Proof: Let Let
be given by
Then, if are distinct, it is impossible that
Proof: With as above, let
be given as follows: Let
be a well-order of
in order type
Let
be the lexicographic ordering on
Set
Lemma 3 There is no
-increasing or decreasing
-sequence of elements of
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Proof: Let be a counterexample. Let
be least such that
has size
and let
be such that if
then
To simplify notation, we will identify
and
For
let
be such that
but
By regularity of
there is
such that
for
many
But if and
then
iff
so
It follows that
has size
contradicting the minimality of
The lemma implies the result: If has size
and is
-homogeneous, then
contradicts Lemma 3.
Now I want to present some significant strengthenings of the results above. The results from last lecture exploit the fact that a great deal of coding can be carried out with infinitely many coordinates. Perhaps surprisingly, strong anti-Ramsey results are possible, even if we restrict ourselves to colorings of pairs.