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 $\omega^2$ into a graph by saying that $(x_0, y_0),(x_1,y_1)\in \omega^2$ form an edge if and only if $|x_0-x_1|+|y_0-y_1|=1$ (that is any point and its direct successor in the product order of $\omega^2$ form an edge).

Is every finite graph an induced subgraph of $\omega^2$?

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 $\omega^2$ into a graph by saying that $(x_0, y_0),(x_1,y_1)\in \omega^2$ form an edge if and only if $|x_0-x_1|+|y_0-y_1|=1$ (that is any point and its direct successor in the product order of $\omega^2$ form an edge).

Is every finite graph an induced minor of $\omega^2$?

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 $\omega^2$ into a graph by saying that $\{(x_0, y_0),(x_1,y_1)\in \omega^2$ form an edge if and only if $|x_0-x_1|+|y_0-y_1|=1$ (that is any point and its direct successor in the product order of $\omega^2$ form an edge).

Is every finite graph an induced subgraph of $\omega^2$?

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 $\omega^2$ into a graph by saying that $(x_0, y_0),(x_1,y_1)\in \omega^2$ form an edge if and only if $|x_0-x_1|+|y_0-y_1|=1$ (that is any point and its direct successor in the product order of $\omega^2$ form an edge).

Is every finite graph an induced subgraph of $\omega^2$?