580 -Partition calculus (7)

May 6, 2009

 
Updates

Let me begin with a couple of updates.

In the last Corollary of the Appendix to lecture I.5, I indicate that in {{\sf ZF},} we have that

\displaystyle \aleph(X)<\aleph({\mathcal P}^2(X))

whenever {\aleph(X)} is not {\aleph_\alpha} for some infinite limit ordinal {\alpha<\aleph_\alpha.} In fact,

\displaystyle {\mathcal P}(\aleph(X))\preceq{\mathcal P}^2(X)

holds.

This result is best possible in terms of positive results. In Theorem 11 of the paper by John Hickman listed at the end, it is shown that for any such {\alpha} it is consistent with {{\sf ZF}} that there is an {X} with {\aleph(X)=\aleph_\alpha} for which {\aleph(X)=\aleph({\mathcal P}^2(X)).}

I also want to give an update on the topics discussed in lecture III.3.

{\mbox{Erd\H os}} and Hajnal asked whether it is possible to have infinite cardinals {\tau,\lambda,\kappa} such that

\displaystyle \tau\rightarrow[\lambda]^\kappa_{\lambda^\kappa}.

Galvin and Prikry showed (see Corollaries 16 and 18 of lecture III.3) that any such {\tau} must be larger than {\lambda^\kappa} and that {\kappa<\lambda.}

Following a nice suggestion of Grigor Sargsyan, we use arguments as in Theorem 9 from lecture III.5 to show that this partition relation cannot hold.

The key is the following:

Lemma 1 If there are infinite cardinals {\tau,\lambda,\kappa} such that {\tau\rightarrow[\lambda]^\kappa_{\lambda^\kappa},} then for every sufficiently large {\gamma} there is an elementary embedding {j:M\rightarrow V_\gamma} such that {|M|=\lambda^\kappa,} {{\rm cp}(j)<\lambda,} {j(\lambda)=\lambda,} and {{}^\kappa M\subseteq M.}

 
Here is a brief sketch:

Proof: By Corollary 20 from lecture III.3, the given relation is equivalent to {\tau\rightarrow[\lambda]^\kappa_\lambda.} Consider a {\kappa}-Skolem function {F:[V_\gamma]^\kappa\rightarrow V_\gamma} so that any {Y\subset V_\gamma} closed under {F} is both closed under {\kappa}-sequences and an elementary substructure of {V_\gamma.}

Use {F} to define a coloring {G:[\tau]^\kappa\rightarrow\lambda} by setting {G(x)=F(x)} whenever {F(x)\in\lambda,} and {G(x)=0} otherwise. By assumption, there is {H\in[\tau]^\lambda} with {G''[H]^\kappa\ne\lambda.} Note that if {Y} is the closure of {H} under {F,} then {Y\cap\lambda=G''[H]^\kappa\cap\lambda\ne\lambda.} But we can assure that {|H\cap\lambda|=\lambda,} and the result follows by taking as {M} the transitive collapse of {H.} \Box

One concludes the proof by noting that it is impossible to have such embeddings. For this, it suffices that {{}^\omega M\subseteq M} and that {M} admits a fixed point past its critical point. One then obtains a contradiction just as in Kunen’s proof that there are no embeddings {j:V\rightarrow V,} see Corollary 9 in lecture III.3.

Similarly, Matthew Foreman has shown that there are no embeddings {j:M\rightarrow V} with {M} closed under {\omega}-sequences. The reason is that any such embedding must admit a fixed point past its critical point, as can be argued from the existence of scales. See the paper by Vickers and Welch listed at the end for a proof of this result.

On the other hand, it is still open whether one can have embeddings {j:M\rightarrow V} such that {M} computes cofinality {\omega} correctly.

1. The Baumgartner-Hajnal theorem

In Theorem 2 of lecture III.5 we showed the {\mbox{Erd\H os}}-Rado result that

\displaystyle \kappa\rightarrow_{top}(Stationary,\omega+1)^2

whenever {\kappa} is regular. It is natural to wonder whether stronger results are possible. We restrict ourselves here to the case {\kappa=\omega_1.} Due to time constraints, we state quite a few results without proof.

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580 -Partition calculus (6)

April 24, 2009

 

1. The {\mbox{Erd\H os}}-Rado theorem

 

Large homogeneous sets (of size {\omega_1} or larger) can be ensured, at the cost of starting with a larger domain. The following famous result was originally shown by {\mbox{Erd\H os}} and Rado using tree arguments (with {\kappa+1} lowered to {\kappa} in the conclusion). We give instead an elementary substructures argument due to Baumgartner, Hajnal and {\mbox{Todor\v cevi\'c},} which proves the stronger version. For {\kappa} a cardinal let {2^{<\kappa}=\sup_{\lambda<\kappa}2^\lambda.}

Theorem 1 ({\mbox{Erd\H os}}-Rado) Let {\kappa} be a regular cardinal and let {\lambda=(2^{<\kappa})^+.} Then

\displaystyle \lambda\rightarrow(\kappa+1)^2_\mu

for all {\mu<\kappa.}

 

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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{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 (Sierpi\’nski) {\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-order 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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580 -Cardinal arithmetic (4)

February 11, 2009

2. Silver’s theorem.

From the results of the previous lectures, we know that any power \kappa^\lambda can be computed from the cofinality and gimel functions (see the Remark at the end of lecture II.2). What we can say about the numbers \gimel(\lambda) varies greatly depending on whether \lambda is regular or not. If \lambda is regular, then \gimel(\lambda)=2^\lambda. As mentioned on lecture II.2, forcing provides us with a great deal of freedom to manipulate the exponential function \kappa\mapsto 2^\kappa, at  least for \kappa regular. In fact, the following holds:

Theorem 1. (Easton). If {\sf GCH} holds, then for any definable function F from the class of infinite cardinals to itself such that:

  1. F(\kappa)\le F(\lambda) whenever \kappa\le\lambda, and
  2. \kappa<{\rm cf}(F(\kappa)) for all \kappa,

there is a class forcing {\mathbb P} that preserves cofinalities and such that in the extension by {\mathbb P} it holds that 2^\kappa=F^V(\kappa) for all regular cardinals \kappa; here, F^V is the function F as computed prior to the forcing extension. \Box

For example, it is consistent that 2^\kappa=\kappa^{++} for all regular cardinals \kappa (as mentioned last lecture, the same result is consistent for all cardinals, as shown by Foreman and Woodin, although their argument is significantly more elaborate that Easton’s). There is almost no limit to the combinations that the theorem allows: We could have 2^\kappa=\kappa^{+16} whenever \kappa=\aleph_\tau is regular and \tau is an even ordinal, and 2^\kappa=\kappa^{+17} whenever \kappa=\aleph_\tau for some odd ordinal \tau. Or, if there is a proper class of weakly inaccessible cardinals (regular cardinals \kappa such that \kappa=\aleph_\kappa) then we could have 2^\kappa= the third weakly inaccessible strictly larger than \kappa, for all regular cardinals \kappa, etc.

Morally, Easton’s theorem says that there is nothing else to say about the gimel function on regular cardinals, and all that is left to be explored is the behavior of \gimel(\lambda) for singular \lambda. In this section we begin this exploration. However, it is perhaps sobering to point out that there are several weaknesses in Easton’s result.

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