Notes about math, research, and more.
If one has a coarse disjoint union of a sequence of finite graphs
\((X_{n})_{n \in \mathbb{N}}\) equipped with the standard graph metric
\(d\) which coarsely embeds into a a sequence of hypercubes \((C_{n})_{n
\in \mathbb{N}}\) equipped with the standard graph metric \(d\), this
does not imply the existence of a hypercubes \((C_{n})_{n \in
\mathbb{N}}\) equipped with the standard graph metric such that the
coarse embedding map embeds all of \(X_{n}\) into \(C_{n}\) for all \(n \in
\mathbb{N}\).
Let \(f:X \to C\) be a coarse embedding. Then we have
\begin{align*} \forall R \geq 0, \exists S \geq 0 \text{ s.t. } d(x,y) \leq R \implies d(f(x), f(y)) \leq S \\ \forall K \geq 0, \exists L \geq 0 \text{ s.t. } d(x,y) \geq K \implies d(f(x), f(y)) \geq L \\ \end{align*}Probably want to do something like \[ {B_{n}} := \sqcup_{n_{1},\ldots,n_{k}}C_{n_{i}} \] where the totality of the image of \(X_{n}\) under \(f\) is contained in \(\cup_{n_{1},\ldots,n_{k}} C_{n_{i}}\).
Then need to show that can uniformly embed the sequence of \((B_{n})_{n \in \mathbb{N}}\) into a new sequence of hypercubes \((\widetilde{C_{n}}}}_{n \in \mathbb{N}}\).