Observations indicate that young massive star clusters in spiral and dwarf galaxies follow a relation between luminosity of the brightest young cluster and the star formation rate (SFR) of the host galaxy, in the sense that higher SFRs lead to the formation of brighter clusters. Assuming that the empirical relation between maximum cluster luminosity and SFR reflects an underlying similar relation between maximum cluster mass (Mecl,max) and SFR, we compare the resulting SFR(Mecl,max) relation with different theoretical models. The empirical correlation is found to suggest that individual star clusters form on a free-fall time-scale with their pre-cluster molecular-cloud-core radii typically being a few parsecs independent of mass. The cloud cores contract by factors of 5-10 while building up the embedded cluster. A theoretical SFR(Mecl,max) relation in very good agreement with the empirical correlation is obtained if the CMF of a young population has a Salpeter exponent of β~ 2.35 and if this cluster population forms within a characteristic time-scale of a 1-10 Myr. This short time-scale can be understood if the interstellar medium is pressurized, thus precipitating rapid local fragmentation and collapse on a galactic scale. Such triggered star formation on a galactic scale is observed to occur in interacting galaxies. With a global SFR of 3-5 Msolar yr-1, the Milky Way appears to lie on the empirical SFR(Mecl,max) relation, given the recent detections of very young clusters with masses near 105 Msolar in the Galactic disc. The observed properties of the stellar population of very massive young clusters suggests that there may exist a fundamental maximum cluster mass, 106 < Mecl,max*/ Msolar< 107.