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Is there an example of an infinite connected, simple, undirected graph $G = (V,E)$ such that every vertex has $|V|$ neighbors, but $G$ does not have a perfect matching (that is, a set $M\subseteq E$ of pairwise disjoint edges such that $\bigcup M = V$)?

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  • $\begingroup$ What’s a perfect matching? $\endgroup$ Apr 23, 2018 at 7:49
  • $\begingroup$ A set $M$ of pairwise disjoint edges such that $\bigcup M = V$ $\endgroup$ Apr 23, 2018 at 8:01
  • $\begingroup$ @BenBarber oh indeed, and it is answered by our student, and I remember now that I have seen this post :) $\endgroup$ Apr 23, 2018 at 14:10

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Consider the ordering on $V$ such that for any vertex $v$ there exist less than $|V|$ vertices $y<x$. Construct the matching inductively.

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