Nuclear collisions can compress nuclear matter to densities achieved within neutron stars and within core-collapse supernovae. These dense states of matter exist momentarily before expanding. We analyzed the flow of matter to extract pressures in excess of 10 34 pascals, the highest recorded under laboratorycontrolled conditions. Using these analyses, we rule out strongly repulsive nuclear equations of state from relativistic mean field theory and weakly repulsive equations of state with phase transitions at densities less than three times that of stable nuclei, but not equations of state softened at higher densities because of a transformation to quark matter.The nucleon-nucleon interaction is generally attractive at nucleon-nucleon separations of 1 to 2 fm (1× 10 -13 cm to 2× 10 -13 cm) but becomes repulsive at small separations ( < 0.5 fm) making nuclear matter difficult to compress. As a consequence, most stable nuclei are at approximately the same "saturation" density, ρ 0 ≈2.7× 10 14 g/cm 3 , in their interiors, and higher densities do not occur naturally on Earth. Matter at densities of up to ρ=9ρ 0 may be present in the interiors of neutron stars (1), and matter at densities up to about ρ=4ρ 0 may be present in the core collapse of type II supernovae (2). The relationship between pressure, density, and temperature described by the equation of state (EOS) of dense matter governs the compression achieved in supernovae and neutron stars as well as their internal structure and many other basic properties (1-5). Models that extrapolate the EOS from the properties of nuclei near their normal density and from nucleon-

Collisions involving112 Sn and 124 Sn nuclei have been simulated with the Improved Quantum Molecular Dynamics transport model. The results of the calculations reproduce isospin diffusion data from two different observables and the ratios of neutron and proton spectra. By comparing these data to calculations performed over a range of symmetry energies at saturation density and different representations of the density dependence of the symmetry energy, constraints on the density dependence of the symmetry energy at sub-normal density are obtained. Results from present work are compared to constraints put forward in other recent analyses.