The effective attraction between like-charged colloids, as induced by multivalent counterions, has been the subject of a variety of computational and theoretical investigations. Nevertheless, the topic is far from controversial, with contradictory predictions and limited experimental observations. A major numerical complication is the dynamic slowdown resulting from the large size asymmetry between the colloids and their counterions. We overcome this problem by extending the generalized geometric cluster algorithm for colloidal suspensions to systems with electrostatic interactions. Using this highly efficient method, we are able to study the onset of like-charged attraction as a function of colloid size. Theoretical considerations for planar geometries predict that the minimum required colloidal charge increases quadratically with colloid size, whereas strong-coupling theory predicts a linear relation. We resolve this controversy by spanning a sufficiently large range in colloidal diameter. Furthermore, we demonstrate important qualitative changes in the potential of mean force as the colloid size is increased.