MathOverflow

The following is motivated by problem number 28 in H. Friedman's 1975 problem list:
For which $S\subseteq\mathbb{N}\cup\{\infty\}$ is there a continuous $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
if $n\in S$ then there is a $g\in C^n\setminus C^{n+1}$ with $(\mathbb{R};<,f)\cong(\mathbb{R};<,g)$, but...

I've asked this question on math.stackexchange a week ago, with no response. More context is available there.
Suppose that $\alpha$ and $\beta$ are closed curves on the $2$-manifold, say $F$ (possibly with boundary), each of which does not disconnect $F$. I want to ensure that the following is true...

Consider the type $A_n$ quiver with gauge group $G=\prod_i \mathrm{GL(V_i)}$ and representation $N=\oplus_i \mathrm{Hom(N_i, N_{i+1})}$, will the K-theoretic Coulomb branch $Spec(\mathrm{K}^{ G(\mathcal{O})}(\mathcal{R}_{G, N}))$ admits a symplectic...

Question: Does there exist $\alpha \colon \mathbb{N} \to \mathbb{N}$ such that the following holds?
If $N$ is a connected smooth $n$-manifold (not necessarily compact), then every two proper smooth embeddings of $N$ in $\mathbb{R}^{\alpha(n)}$ are smoothly ambient isotopic....