For any graph $G=(V,E)$ we define $\mu(G) = \sup\{|M|: M\subseteq E(G) \text{ is a matching}\}$.


Is there a graph $G=(V,E)$ such that for every matching $M\subseteq E$ we have $|M|<\mu(G)$?



Maximum matchings in infinite graphs
Yes. First start off with the disjoint union $G=\bigcup\_{n\in\mathbb{N}}K\_n$ where $K\_n$ is the complete graph on $\{1,\ldots,n\}$ for $n\in\mathbb{N}, n\geq 1$.


Note that this graph contains cliques of size $n$ for every $n\in\mathbb{N}$, but it does not contain a clique of size $\omega$. 


Let $G^c$ be the complement of the graph $G$ constructed above. If we define its independence number to be $\alpha(G^c) = \sup\{|I|: I\subseteq V(G^c) \text{ and } I \text { is independent}\}$.


Clearly, every independent set of $G^c$ is finite, but $\alpha(G^c)=\omega$.


Finally we consider the [line graph](http://en.wikipedia.org/wiki/Line_graph) $L(G^c)$. Note that we have a correspondence of independent sets in $G^c$ and matchings in $L(G^c)$. Therefore $L(G^c)$ contains only finite matchings, but $\mu(L(G^c)) = \alpha(G^c) = \omega$.


