Some questions about the order on the hyperreals and the Ramsey theory on it.

Question

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.

Answer 1

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.