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To constrain for the first time the mean mass and extent of the molecular gasof a mass-complete sample of $>10^{10}$M$_{\odot}$ main-sequence (MS) galaxiesat $0.4<z<3.6$. We apply an innovative $uv$-based stacking analysis to a largeset of archival Atacama Large Millimeter/submillimeter Array (ALMA)observations. This stacking analysis provides measurements of the mean mass andextent of the molecular gas of galaxy populations. The molecular gas mass of MSgalaxies evolves with redshift and stellar mass. At all stellar masses, themolecular gas fraction decreases by a factor of 24 from $z\sim3.2$ to $z\sim0$.At a given redshift, the molecular gas fraction of MS galaxies decreases withstellar mass, at roughly the same rate as their specific star formation ratedecreases. The molecular gas depletion time of MS galaxies remains roughlyconstant at $z>0.5$ with a value of 300--500 Myr, but increases by a factor of3 from $z\sim0.5$ to $z\sim0$. This evolution of the molecular gas depletiontime of MS galaxies can be predicted from the evolution of their molecular gassurface density and a seemingly universal MS-only $\Sigma_{M_{\rmmol}}-\Sigma_{\rm SFR}$ relation with an inferred slope of 1.13, i.e., theso-called KS relation. The far-infrared size of MS galaxies shows nosignificant evolution with redshift or stellar mass, with a mean circularizedhalf-light radius of 2.2 kpc. Finally, our mean molecular gas masses are lowerthan previous estimates, likely caused by the fact that literature studies werebiased towards individually-detected MS galaxies with massive gas reservoirs.To first order, the molecular gas content of MS galaxies regulates their starformation across cosmic time, while variation of their star formationefficiency plays a secondary role. Despite a large evolution of their gascontent and SFRs, MS galaxies evolved along a seemingly universal MS-only KSrelation.

A$^{3}$COSMOS: A census on the molecular gas mass and extent of main-sequence galaxies across cosmic time