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,
Proof: Let
Let
be given by

Then, if
are distinct, it is impossible that

Theorem 2 (Sierpiński)
In fact,
Proof: With
as above, let
be given as follows: Let
be a well-ordering of
in order type
Let
be the lexicographic ordering on
Set

Lemma 3 There is no
-increasing or decreasing
-sequence of elements of
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.
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