To the best of my knowledge, stiffness of ordinary differential equations is difficult to capture but can be roughly described as problems where explicit methods don't work while implicit ones do. Alternatively, the stiffness ratio of the Jacobian of the dynamics (i.e. the ratio of its largest to smallest eigenvalues) is supposed to be a quantitative indicator of a problem's stiffness. The advantage of implicit over explicit methods seems to be that they remain stable for larger time steps, even in the face of stiffness.
Does that mean that stiffness and instability are fundamentally linked? As stability is a property of the solver and stiffness a property of the problem, I would think that would not be the case. Are there any problems that are not stiff (e.g. have a small stiffness ratio) but explicit methods are still unstable on them? On the flip side, are there any problems with large stiffness that can be solved with explicit methods?