## 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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## 305 -Homework set 9

May 2, 2009

[Update: I added an additional assumption to question 3.]

This set is due May 13 at 10:30 am. Remember that we will have an additional meeting that day. Details of the homework policy can be found on the syllabus and here. This set is extra credit.

Let ${X}$ be a finite set and let ${G}$ be a subgroup of the group ${S_X}$ of permutations of the elements of ${X.}$ Define a relation ${\sim}$ on ${X}$ by saying that ${x\sim y}$ iff either ${x=y}$ or ${(x,y)\in G.}$

• Begin by showing that ${\sim}$ is an equivalence relation.

For each ${x\in X,}$ denote by ${E_x}$ the equivalence class of ${x,}$ i.e.,

$\displaystyle E_x=\{y\in X : x\sim y\}.$

Suppose now that in addition, for any ${x,y\in X}$ there is some permutation ${\pi\in G}$ such that ${\pi(x)=y.}$

• Show that all the equivalence classes have the same size: ${|E_x|=|E_y|}$ for all ${x,y.}$

Now assume as well that ${|X|=p}$ is a prime number, and that $G$ contains at least one transposition ${(x,z)}$ for some ${x,z\in X}$ with ${x\ne z.}$

• Conclude that ${G=S_X.}$

As an application, suppose that ${p}$ is a prime number and ${f\in{\mathbb Q}[x]}$ is irreducible and of degree ${p.}$ Assume that ${f}$ has exactly ${p-2}$ real roots and 2 complex (non-real) roots.

• Conclude that

$\displaystyle {\rm Gal}({\mathbb Q}^{f(x)}/{\mathbb Q})\cong S_p.$

In particular, let ${f(x)=x^5-4x+2.}$

• Show that ${{\rm Gal}({\mathbb Q}^{f(x)}/{\mathbb Q})\cong S_5.}$

Now suppose that ${2r+3}$ is prime, and let

$\displaystyle f_r(x)=(x^2+4)x(x^2-4)(x^2-16)\dots(x^2-4r^2).$

• Show that if ${k}$ is an odd integer then ${|f_r(k)|\ge5.}$ Let ${g(x)=f_r(x)-2,}$ and show that ${g}$ is irreducible over ${{\mathbb Q}.}$ Now find ${{\rm Gal}({\mathbb Q}^{g(x)}/{\mathbb Q}).}$

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