Let $G$ be an undirected graph with an infinite number of vertices (and edges), and assume it is connected in the sense every $u,v$ have at least one path connecting them. Assume each vertex has a finite degree. My question is: Is it possible for a pair of vertices to exist, such that they will have an infinite number of paths?
Ofcourse, if we allow a path to contain a cycle, the answer is positive. However, I am interested in what happens when we forbid paths that contain cycles.
In other words: Is it possible for a pair of vertices to exist, such that they will have an infinite amount of simple paths?