In this work we explore the kinetics of single-crystal graphene growth as a function of nucleation density. In addition to the standard methods for suppressing nucleation of graphene by pretreatment of Cu foils using oxidation, annealing, and reduction of the Cu foils prior to growth, we introduce a new method that further reduces the graphene nucleation density by interacting directly with the growth process at the onset of nucleation. The successive application of these two methods results in roughly 3 orders of magnitude reduction in graphene nucleation density. We use a kinetic model to show that at vanishingly low nucleation densities carbon incorporation occurs by a cooperative island growth mechanism that favors the formation of substrate-size single-crystal graphene. The model reveals that the cooperative growth of millimeter-size single-crystal graphene grains occurs by roughly 3 orders of magnitude increase in the reactive sticking probability of methane compared to that in random island nucleation.