Finding subgraphs with high treewidth and constant degree I am given a graph $G$ with treewidth $k$ and arbitrary degree, and I would like to find a subgraph $H$ of $G$ (not necessarily an induced subgraph) such that $H$ has constant degree and its treewidth is as high as possible. Formally my problem is the following: having chosen a degree bound $d \in \mathbb{N}$, what is the "best" function $f : \mathbb{N} \to \mathbb{N}$ such that, in any graph $G$ with treewidth $k$, I can find (hopefully efficiently) a subgraph $H$ of $G$ with maximal degree $\leq d$ and treewidth $f(k)$.

Obviously we should take $d \geq 3$ as there are no high treewidth graphs with maximal degree $&lt;3$. For $d = 3$ I know that you can take $f$ such that $f(k) = \Omega(k^{1/100})$ or so, by appealing to Chekuri and Chuzhoy's grid minor extraction result (and using it to extract a high-treewidth degree-3 graph, e.g., a wall, as a topological minor), with the computation of the subgraph being feasible (in RP). However, this is a very powerful result with an elaborate proof, so it feels wrong to use it for what looks like a much simpler problem: I would just like to find any constant-degree, high-treewidth subgraph, not a specific one like in their result. Further, the bound on $f$ is not as good as I would have hoped. Sure, it is known that it can be made $\Omega(k^{1/20})$ (up to giving up efficiency of the computation), but I would hope for something like $\Omega(k)$. So, is it possible to show that, given a graph $G$ of treewidth $k$, there is a subgraph of $G$ with constant degree and linear treewidth in $k$?

I'm also interested in the exact same question for pathwidth rather than treewidth. For pathwidth I don't know any analogue to grid minor extraction, so the problem seems even more mysterious...
See the paper by Julia Chuzhoy and myself on Treewidth sparsifiers. 
We show that one can obtain a subgraph of degree at most 3 with treewidth $\Omega(k/polylog(k))$ where $k$ is the treewidth of $G$. 
https://arxiv.org/abs/1410.1016 The proof is shorter than the one 
for grid minors but it is still not that that easy and builds on several previous 
tools.

Suppose you settle for an easier target - degree 4 and treewidth $\Omega(k^{1/4})$ then you can get it much more easily via  result of Reed and Wood on grid-like minors. https://arxiv.org/abs/0809.0724

Another easy result you can obtain is the following which is a starting point for some of the more involved proofs. You can get a subgraph of degre  $\log^2(k)$ 
and treewidth $\Omega(k/\mathsf{polylog}(k))$. You can see the treewidth sparsifier paper for the argument to achieve this.