We present the atmospheric structure and the fundamental properties of the red supergiants (RSGs) AH Sco, UY Sct, and KW Sgr based on VLTI/AMBER observations. We carried out spectro-interferometric observations of AH Sco, UY Sct, and KW Sgr in the near-infrared K band with the VLTI/AMBER instrument, and compared the data to a new grid of hydrostatic PHOENIX model atmospheres. In our visibility data, we observe molecular layers of water and CO in extended atmospheres. For a uniform disk modeling, we observe size increases at the water band of 10% to 25% and at the CO bandheads of 20%-35% with respect to the near continuum bandpass. The PHOENIX atmosphere models predict the spectra and the continuum visibility values, but cannot reproduce the large extensions of the molecular layers. This indicates that the opacities of the molecular bands are included, but that the model atmospheres are too compact compared to the observations. The observed extended layers may be levitated by processes such as pulsation or convection, which are not included in the hydrostatic atmospheric models. Comparing the continuum visibility values to PHOENIX models, we estimate the Rosseland-mean photospheric angular diameters. Together with the distance and the spectro-photometry, we calculate radii and effective temperatures. The location of the targets in the HR-diagram is confirmed to be close to, and possibly slightly to the right of, the Hayashi limit of recent evolutionary tracks corresponding to masses between about 20 Msun and 40 Msun
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