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 $\{0,1\}^\omega$?

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    $\begingroup$ I think the term "induced minor" is more commonly used for what you are referring to: an induced minor is obtained from a graph by vertex deletion and edge contraction. Is that correct? $\endgroup$ – Puck Rombach Jan 1 at 17:11
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    $\begingroup$ $2^\omega$ consists of continuum many paiwise isomorphic components. Each component is countable. Hence every connected minor is countable. (I think I gave a similar answer earlier: mathoverflow.net/questions/301942/… ) $\endgroup$ – Goldstern Jan 1 at 17:45
  • $\begingroup$ Right @PuckRombach, will correct the terminology. $\endgroup$ – Dominic van der Zypen Jan 1 at 20:27
  • $\begingroup$ Thanks @Goldstern, have removed the question for uncountable minors $\endgroup$ – Dominic van der Zypen Jan 1 at 20:29

Let $G=(\omega,E)$ be an arbitrary graph. Let $S_n=\{x_n\}\cup\{y_{nm}:\{n,m\}\in E\}$, where $x_n$ is the characteristic function of $\{2n\}$ and $y_{nm}$ is the characteristic function of $\{2m,2n+1\}$. The induced minor of $\mathcal S=\{S_n:n\in\omega\}$ is clearly isomorphic to $G$, showing every countable graph is an induced minor. (Note: in the definition of $S_n$ we could even restrict $m<n$, which would let each $S_n$ be finite.)

As Goldstern remarks in a comment, every connected component of $\{0,1\}^\omega$ is countable, hence so is every connected minor. Since the connected components are isomorphic and there is $2^{\aleph_0}$ of them, we can conclude induced minors of this graph are precisely ones which are unions of at most continuum many countable graphs.


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