Ramsey theory of very small countable ordinals

November 6, 2014

I was an undergraduate student at Los Andes, from 1992 to 1996. This year, their mathematics program is turning 50. There was a conference in September to celebrate the event, and I had the honor to give one of the talks (see here for the English version of the slides).

The Faculty of Science publishes a magazine, Hipótesis, and a special edition will be devoted to the conference. I have submitted an expository paper, based on my talk.

The topic is the partition calculus of very small countable ordinals (mainly ordinals below \omega^2, actually). The paper reviews Ramsey’s theorem and a few finite examples, before discussing the two main results.

1.

One is an old theorem by Haddad and Sabbagh, unfortunately not well known. In 1969, they computed the Ramsey numbers r(\omega+n,m) for n,m finite.

Given nonzero ordinals \alpha,\beta, recall that r(\alpha,\beta) is the least \gamma such that any red-blue coloring of the edges of K_\gamma either admits a red copy of K_{\alpha} or a blue copy of K_\beta. Clearly r(\alpha,1)=1, r(\alpha,\beta)\ge r(\alpha,2)=\alpha if \beta\ge2, and r(\alpha,\beta)=r(\beta,\alpha), so we may as well assume that \alpha\ge\beta>2, and we adopt this convention in what follows.

Ramsey proved two theorems about this function in a famous 1928 paper that introduced the topic. In the notation we have just set up, his first result asserts that r(n,m) is finite whenever n,m are finite, and his second result states that r(\omega,\omega)=\omega. The computation of the numbers r(n,m) is an extremely difficult, most likely unfeasible, problem, though r is obviously a recursive function. We are concerned here with the values of the function when at least one of the arguments is infinite.

It turns out that r(\omega+1,\omega) is already \omega_1. Hence, if we are interested in studying the countable values of the function r(\alpha,\beta), then we must assume that either \omega=\alpha, in which case r(\alpha,\beta)=\omega and there is nothing more to say, or else (that is, if \alpha is countable and strictly larger than \omega) we must assume that \beta is finite.

The function has been intensively studied when \alpha is a limit ordinal, particularly a power of \omega. Here we look at the much humbler setting where \omega<\alpha<\omega2. Recalling that each ordinal equals the set of its predecessors, and using interval notation to describe sets of ordinals, the Haddad-Sabbagh result is as follows:

Lemma. For all positive integers n, m there exists a positive integer k\ge n, m such that for any red-blue coloring of the edges of K_{[0,k)}, and such that K_{[0,m)} is blue, there is a subset H of {}[0, k) with K_H monochromatic, and either K_H is blue and |H| = m + 1, or else K_H is red, |H|=n+1, and H\cap[0,m)\ne\emptyset.

Denote by r_{HS}(n+1,m+1) the smallest number k witnessing the lemma.

Theorem. If n,m are positive integers, then r(\omega +n,m)=\omega(m-1)+t, where r_{HS}(n+1,m)=(m-1)+t.

The theorem was announced in 1969, but the proof never appeared. I have written a survey on the topic, including what should be the first proof in print of this result.

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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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