Let $G=(V,E)$ be a simple, undirected graph. Suppose that ${\cal S}$ is a collection of non-empty, connected, and pairwise disjoint subsets of $V$. Let $G({\cal S})$ be the graph with vertex set ${\cal S}$; and $S\neq T\in {\cal S}$ form an edge if and only if if there are $x\in S, y \in T$ such that $\{x,y\}\in E$.

If $H$ is a simple undirected graph, we say that $H$ is a *induced minor* of $G$ if there is a collection ${\cal S}$ of non-empty, connected, and pairwise disjoint subsets of $V(G)$ such that $H\cong G({\cal S})$.

We make $\{0,1\}^\omega$ into a graph by saying that $x,y\in\{0,1\}^\omega$ form an edge if $|\{k\in\omega:x(k)\neq y(k)\}|=1$.

Is every countable graph a induced minor of $\{0,1\}^\omega$?