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
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
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.