  $Kepler$ revealed that roughly one-third of Sun-like stars host planets
orbiting within 100 days and between the size of Earth and Neptune. How do
these planets form, what are they made of, and do they represent a continuous
population or multiple populations? To help address these questions, we began
the Magellan-TESS Survey (MTS), which uses Magellan II/PFS to obtain radial
velocity (RV) masses of 30 TESS-detected exoplanets and develops an analysis
framework that connects observed planet distributions to underlying
populations. In the past, small planet RV measurements have been challenging to
obtain due to host star faintness and low RV semi-amplitudes, and challenging
to interpret due to the potential biases in target selection and observation
planning decisions. The MTS attempts to minimize these biases by focusing on
bright TESS targets and employing a quantitative selection function and
observing strategy. In this paper, we (1) describe our motivation and survey
strategy, (2) present our first catalog of planet density constraints for 27
TESS Objects of Interest (TOIs; 22 in our population analysis sample, 12 that
are members of the same systems), and (3) employ a hierarchical Bayesian model
to produce preliminary constraints on the mass-radius (M-R) relation. We find
that the biases causing previous M-R relations to predict fairly high masses at
$1~R_\oplus$ have been reduced. This work can inform more detailed studies of
individual systems and offer a framework that can be applied to future RV
surveys with the goal of population inferences.
