Is there a countable, simple, connected graph $G=(\omega, E)$ such that $\text{deg}(v)$ is infinite for all $v\in \omega$, and for all [matchings](https://en.wikipedia.org/wiki/Matching_(graph_theory)) $M\subseteq E$ the set $V\setminus (\bigcup M)$ is infinite?



Connected infinite graphs in which all matchings are "small"
If all degrees are infinite, it contains a perfect matching: just add edges one by one covering all vertices.


