Suppose $G=(V,E)$ is a simple, undirected graph with $|V|,|E|$ infinite. Is there $B\subseteq E$ with $|B| = |E|$ such that $(V,B)$ is bipartite?



Bipartite subgraphs with lots of edges
Yes. In each component Take a vertex $v$ and for every vertex $u$ let $d(v,u)$ be the shortest distance in $G$ between $v$ and $u$, where distance is defined as the fewest number of edges. Then every edge $\{u,u'\}$ in $G$ satisfies $|d(v,u)-d(v,u')| \le 1$, and the graph formed from $G$ by removing all edges between vertices $u$ and $u'$ s.t $d(v,u')=d(v,u)$ is bipartite. 


