The paper by Akmal, Pandharipande, and Ravenhall investigates the properties of dense nucleon matter and the structure of neutron stars using variational chain summation methods and the Argonne $v_{18}$ two-nucleon interaction. The study includes relativistic boost corrections to the two-nucleon interaction and three-nucleon interactions, which increase the mass limit of stable neutron stars from 1.67 to 2.20 solar masses ($M_{\odot}$). The Hamiltonians predict a transition to a phase with neutral pion condensation at a baryon number density of $\sim 0.2$ fm$^{-3}$. Neutron stars are found to have a thin layer (tens of meters thick) where the density changes rapidly from normal to the condensed phase. The paper also explores the possibility of dense nucleon matter containing quark matter, using the bag model equation of state. It is found that pure neutron stars do not have quark matter admixtures, but heavier stars may have cores consisting of a mixture of quark and nucleon matter, reducing the maximum mass to 2.02 ($1.91$) $M_{\odot}$ for bag constants $B = 200$ ($122$) MeV/fm$^3$. The authors also consider the possibility of maximally incompressible matter above a certain density, which limits the maximum mass of neutron stars to below $2.5M_{\odot}$. The effects of phase transitions on the composition and adiabatic index of neutron star matter are discussed.The paper by Akmal, Pandharipande, and Ravenhall investigates the properties of dense nucleon matter and the structure of neutron stars using variational chain summation methods and the Argonne $v_{18}$ two-nucleon interaction. The study includes relativistic boost corrections to the two-nucleon interaction and three-nucleon interactions, which increase the mass limit of stable neutron stars from 1.67 to 2.20 solar masses ($M_{\odot}$). The Hamiltonians predict a transition to a phase with neutral pion condensation at a baryon number density of $\sim 0.2$ fm$^{-3}$. Neutron stars are found to have a thin layer (tens of meters thick) where the density changes rapidly from normal to the condensed phase. The paper also explores the possibility of dense nucleon matter containing quark matter, using the bag model equation of state. It is found that pure neutron stars do not have quark matter admixtures, but heavier stars may have cores consisting of a mixture of quark and nucleon matter, reducing the maximum mass to 2.02 ($1.91$) $M_{\odot}$ for bag constants $B = 200$ ($122$) MeV/fm$^3$. The authors also consider the possibility of maximally incompressible matter above a certain density, which limits the maximum mass of neutron stars to below $2.5M_{\odot}$. The effects of phase transitions on the composition and adiabatic index of neutron star matter are discussed.