For any set $X$, let $[X]^2=\{\{x,y\}: x\neq y\in X\}$.

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. Note that, thanks to @Wojowu's comment below, the following holds. If $f, g\in \omega^\omega$ have the property that $g\circ f$ has a fixed point, then so does $f\circ g$. So, let $$E = \big\{\{f,g\}\in[\omega^\omega]^2: \exists n\in\omega\big((g\circ f)(n)=n\big)\big\}.$$ Then $(\omega^\omega,E)$ is a simple, undirected graph.

Is every finite simple, undirecte graph isomorphic to some induced subgraph of $(\omega^\omega,E)$? What if we consider graphs with countable vertex set?

For every set $X$, let $[X]^2=\{\{x,y\}: x\neq y\in X\}$.

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. Note that, thanks to @Wojowu's comment below, the following holds. If $f, g\in \omega^\omega$ have the property that $g\circ f$ has a fixed point, then so does $f\circ g$. So, let $$E = \bigl\{\{f,g\}\in[\omega^\omega]^2: \exists n\in\omega\bigl((g\circ f)(n)=n\bigr)\bigr\}.$$ Then $(\omega^\omega,E)$ is a simple, undirected graph.

Is every finite simple, undirected graph isomorphic to some induced subgraph of $(\omega^\omega,E)$? What if we consider graphs with countable vertex set?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. Let $$E = \big\{(f,g)\in(\omega^\omega)^2: \exists n\in\omega\big((g\circ f)(n)=n\big)\big\}.$$ So $(\omega^\omega,E)$ is a directed graph (with lots of self-loops).

Question. Is every finite directed graph isomorphic to some induced subgraph of $(\omega^\omega,E)$?

For any set $X$, let $[X]^2=\{\{x,y\}: x\neq y\in X\}$.

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. Note that, thanks to @Wojowu's comment below, the following holds. If $f, g\in \omega^\omega$ have the property that $g\circ f$ has a fixed point, then so does $f\circ g$. So, let $$E = \big\{\{f,g\}\in[\omega^\omega]^2: \exists n\in\omega\big((g\circ f)(n)=n\big)\big\}.$$ Then $(\omega^\omega,E)$ is a simple, undirected graph.

Is every finite simple, undirecte graph isomorphic to some induced subgraph of $(\omega^\omega,E)$? What if we consider graphs with countable vertex set?