There are now many examples of integer homology spheres which cannot be written as surgery on a knot, but examples which cannot be surgery on some 2-component link have remained out of reach. From one perspective, the difficulty is that the trace of the surgery is an indefinite 4-manifold, and many of our favorite invariants provide no (or limited) constraints on indefinite boundings.I will describe the structure of the mod-2 instanton complex, and Froyshov's invariant q_3(Y) in Z for an integer homology sphere. If W: Y -> Y' is a cobordism with H_1(W; Z/2) = 0, this satisfies -b^+(W) <= q_3(Y') - q_3(Y) <= b^-(W). I will sketch the origin of this inequality. A computation then gives examples of hyperbolic 3-manifolds Y_n which are integral Dehn surgery on an n-component link, but not rational Dehn surgery on any link of fewer than n components