580 -Partition calculus (4)

April 9, 2009

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 ({\mbox{\bf Erd\H os}}-Kakutani) {\omega_1\not\rightarrow(3)^2_\omega.} In fact, {2^\kappa\not\rightarrow(3)^2_\kappa.}

Proof: Let {S={}^\kappa\{0,1\}.} Let {F:[S]^2\rightarrow\kappa} be given by

\displaystyle F(\{f,g\})=\mbox{least }\alpha<\kappa\mbox{ such that }f(\alpha)\ne g(\alpha).

Then, if {f,g,h} are distinct, it is impossible that {F(\{f,g\})=F(\{f,h\})=F(\{g,h\}).} \Box

Theorem 2 (Sierpiński) {\omega_1\not\rightarrow(\omega_1)^2.} In fact, {2^\kappa\not\rightarrow(\kappa^+)^2.}

Proof: With {S} as above, let {F:[S]^2\rightarrow2} be given as follows: Let {<} be a well-ordering of {S} in order type {2^\kappa.} Let {<_{lex}} be the lexicographic ordering on {S.} Set

\displaystyle F(\{f,g\})=1\mbox{ iff }<_{lex}\mbox{ and }<\mbox{ coincide on }\{f,g\}.

Lemma 3 There is no {<_{lex}}-increasing or decreasing {\kappa^+}-sequence of elements of {S.}

Proof: Let {W=\{f_\alpha\colon\alpha<\kappa^+\}} be a counterexample. Let {\gamma\le\kappa} be least such that {\{f_\alpha\upharpoonright\gamma\colon\alpha<\kappa^+\}} has size {\kappa^+,} and let {Z\in[W]^{\kappa^+}} be such that if {f,g\in Z} then {f\upharpoonright\gamma\ne g\upharpoonright\gamma.} To simplify notation, we will identify {Z} and {W.} For {\alpha<\kappa^+} let {\xi_\alpha<\gamma} be such that {f_\alpha\upharpoonright\xi_\alpha=f_{\alpha+1} \upharpoonright\xi_\alpha} but {f_\alpha(\xi_\alpha)=1-f_{\alpha+1}(\xi_\alpha).} By regularity of {\kappa^+,} there is {\xi<\gamma} such that {\xi=\xi_\alpha} for {\kappa^+} many {\alpha.}

But if {\xi=\xi_\alpha=\xi_\beta} and {f_\alpha\upharpoonright\xi=f_\beta\upharpoonright\xi,} then {f_\beta<_{lex} f_{\alpha+1}} iff {f_\alpha<_{lex} f_{\beta+1},} so {f_\alpha=f_\beta.} It follows that {\{f_\alpha\upharpoonright\xi\colon\alpha<\kappa^+\}} has size {\kappa^+,} contradicting the minimality of {\gamma.} \Box

The lemma implies the result: If {H\subseteq S} has size {\kappa^+} and is {F}-homogeneous, then {H} contradicts Lemma 3. \Box

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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