We show that the Uemura relation, namely Tc{T}_{c} proportional to superfluid density \ensuremathρs,{\ensuremath{\rho}}_{s}, is not generally satisfied in high-Tc{T}_{c} superconductors. Instead, a modified plot of Tc/\ensuremathΔ0{T}_{c}/{\ensuremath{\Delta}}_{0} versus \ensuremathρs{\ensuremath{\rho}}_{s} where \ensuremathΔ0{\ensuremath{\Delta}}_{0} is the maximum d-wave gap at T=0,T=0, exhibits universal features that point to an alternative interpretation of the underlying physics. In the underdoped region this plot exhibits the canonical negative curvature expected when a ground-state correlation competes with superconductivity (SC) by opening up a gap (the pseudogap) in the normal-state density of states. Moreover we observe simple linear relations between Tc\ensuremathρs,{T}_{c}{\ensuremath{\rho}}_{s}, the electronic entropy at Tc,{T}_{c}, and the critical impurity concentration for suppressing SC. These relations confirm that the pseudogap persists to T=0,T=0, strongly modifying the SC ground state.