Black disk radius constrained by unitarity

We argue that if the elastic proton–proton cross section increases with energy, the Froissart-like high energy behaviour of the elastic amplitude (which corresponds to a 'black disk' of radius $R\left(s\right)=c\ln s-\beta \ln (\ln s)$) is the only possibility to satisfy the unitarity equation at each value of the impact parameter, b. Otherwise the cross section of events with Large Rapidity Gaps grows faster than the total cross section at the same b. That is, these 'gap' events require maximal growth of the high-energy (asymptotic) cross section and of the interaction radius $R\left(s\right)$ in order to be consistent with unitarity.