Given a finite connected graph, let $A$ be a set of edges such that each edge in $A$ is not part of a cycle. Suppose that no path contains all edges in $A$. Must it be true that for some three edges in $A$, no path contains all the three edges?
This is equivalent to showing that if every subset of three edges in $A$ is contained in a path, then there is a path containing all edges in $A$. I’ve tried doing induction, but even going from three to four doesn’t seem obvious. Each set of edges $\{e_1,e_2,e_3\}$ and $\{e_2,e_3,e_4\}$ has a path containing it, but other edges maybe used in both of these paths in various orders.