Graphs with cycle spaces generated by bounded-length cycles have the coarse Menger property, with corollaries for hyperbolic graphs, finitely presented groups, and planar graphs with bounded faces.
A counterexample to the coarse Menger conjecture
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A coarse Menger theorem for hyperbolic graphs, finitely presented groups, and more
Graphs with cycle spaces generated by bounded-length cycles have the coarse Menger property, with corollaries for hyperbolic graphs, finitely presented groups, and planar graphs with bounded faces.