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We present the detection of multiple CO line transitions with ALMA in a fewtens of infrared-selected galaxies on and above the main sequence at z=1.1-1.7.We reliably detected the emission of CO(5-4), CO(2-1), and CO(7-6)+[CI](2-1) in50, 33, and 13 galaxies, respectively, and we complemented this informationwith available CO(4-3) and [CI](1-0) fluxes for part of the sample, andmodeling of the optical-to-mm SEDs. We retrieve a quasi-linear relation betweenLIR and CO(5-4) or CO(7-6) for main-sequence galaxies and starbursts,corroborating the hypothesis that these transitions can be used as SFR tracers.We find the CO excitation to steadily increase as a function of the starformation efficiency, the mean intensity of the radiation field warming thedust, the surface density of SFR, and, less distinctly, with the distance fromthe main sequence. This adds to the tentative evidence for higher excitation ofthe CO+[CI] SLED of starbursts relative to that for main-sequence objects,where the dust opacities play a minor role in shaping the high-J CO transitionsin our sample. However, the distinction between the average SLED of uppermain-sequence and starburst galaxies is blurred, driven by a wide variety ofintrinsic shapes. LVG radiative transfer modeling demonstrates the existence ofa highly excited component that elevates the CO SLED of high-redshiftmain-sequence and starbursting galaxies above the typical values observed inthe disk of the Milky Way. This excited component is dense and it encloses ~50%of the total molecular gas mass in main-sequence objects. We interpret theobserved trends involving the CO excitation as mainly driven by a combinationof large SFRs and compact sizes, naturally connected with enhanced densemolecular gas fractions and higher dust and gas temperatures, due to increasingUV radiation fields, cosmic ray rates, and dust/gas coupling. [Abridged]

CO emission in distant galaxies on and above the main sequence