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Let $^* \mathbb R$ denote the set of hyperreal numbers, which is constructed with a nonprincipal ultrafilter on $\mathbb N$. Since we can order-embed $\omega_1$ into $^* \mathbb R$, by the Erdős-Dushnik-Miller theorem, we have $^* \mathbb R \rightarrow (\omega_1, \omega)^2$. I have $3$ questions regarding the order on hyperreal numbers and the Ramsey theory on it:

  1. What order-types can we realize in $^*\mathbb R$? (I know that this question is not very precise but, for example, $\omega + (\omega^* + \omega) \cdot \eta$ is realized in any nonstandard model of arithmetic, where $\omega^*$ denotes the converse of $\omega$, and $\eta$ is the order-type of the rational numbers with the usual ordering, so I am expecting an answer like this but for $^*\mathbb R$).

  2. Can we order-embed any ordinal $\alpha < \omega_2$ into $^* \mathbb R$? Note that since any hyperreal interval $(a,b)$ with real endpoints is order-isomorphic to $^* \mathbb R$, we can extract countably many $\omega_1$’s from $^*\mathbb R$, but can we do better than that?

  3. Finally, for what ordinals $\alpha < \omega_2$ do we have $^* \mathbb R \rightarrow (\alpha, \omega)^2$? For the simplest case, do we have $^* \mathbb R \rightarrow (\omega_1 + 1, \omega)^2$? Or even $^* \mathbb R \rightarrow (\omega_1+1, 3)^2$? (Note that $\omega_1 + 1$ order-embeds into $^* \mathbb R$ by (2).)

It can also be interesting to see if we can further generalize (3) by increasing the second index from $\omega$ to some bigger countable ordinal $\alpha$ (note that it cannot be uncountable since $2^{\aleph_0} \nrightarrow (\aleph_1, \aleph_1)^2)$, thank you.

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I don't know about the Ramsey theory, but your first two questions have easy answers.

Since $^*\mathbb{R}$ is $\aleph_1$-saturated, every linear order $L$ of cardinality at most $\aleph_1$ embeds in $^*\mathbb{R}$. In particular, every ordinal $<\omega_2$ embeds in $^*\mathbb{R}$.

Proof: Enumerate $L$ by $\omega_1$. Now build the embedding one element at a time by transfinite induction. At each successor step, we only have to fill cuts in countable subsets of $^*\mathbb{R}$, which we can always do.

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  • $\begingroup$ Can we answer the first question with a “nice” form of an order-type? $\endgroup$
    – Tan
    Jan 22 at 14:04
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    $\begingroup$ @Tan I'm not sure what you mean. Name any order type you like of cardinality at most $\aleph_1$. It embeds. $\endgroup$ Jan 22 at 14:05
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    $\begingroup$ @Tan Note that the same result (any linear order of size $\aleph_1$ embeds) holds for ultrapowers of the natural numbers. The weaker result about $\omega+(\omega*+\omega)\cdot \eta$ holds for arbitrary nonstandard models of arithmetic (which might be countable). Similarly, since the hyperreals are an ultrapower, we get $\aleph_1$-saturation. This is not true for arbitrary nonstandard models of the theory of the real numbers. $\endgroup$ Jan 22 at 14:23
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    $\begingroup$ @Tan Certainly. There are countable models of this theory, like the real algebraic numbers. But maybe by "nonstandard model", you mean a proper elementary extension of the reals. Then consider, for example, the field of Puiseux series over the real numbers. This field still has a countable subset which is dense in the order topology (the Puiseux monomials with coefficients in the rationals), so it fails to embed $\omega_1$. $\endgroup$ Jan 22 at 15:28
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    $\begingroup$ Thank you, your answer is really helpful but I will wait to see if anyone can answer the Ramsey theory part. $\endgroup$
    – Tan
    Jan 22 at 15:52

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