  High energy heavy-ion collisions are proposed as a novel means to obtain
information about the high density ({\rm HD}) behaviour of nuclear symmetry
energy. Within an isospin-dependent hadronic transport model using
phenomenological equations of state ({\rm EOS}) for dense neutron-rich matter,
it is shown that the isospin asymmetry of the HD nuclear matter formed in high
energy heavy-ion collisions is determined mainly by the HD behaviour of nuclear
symmetry energy. Experimental signatures in several sensitive probes, i.e.,
$\pi^-$ to $\pi^+$ ratio, transverse collective flow and its excitation
function as well as neutron-proton differential flow, are investigated. A
precursor of the possible isospin separation instability in dense neutron-rich
matter is predicted to appear as the local minima in the excitation functions
of the transverse flow parameter for both neutrons and protons above the pion
production threshold. Because of its {\it qualitative} nature unlike other {\it
quantitative} observables, this precursor can be used as a unique signature of
the isospin dependence of the nuclear {\rm EOS}. Measurements of these
observables will provide the first terrestrial data to constrain stringently
the HD behaviour of nuclear symmetry energy and thus also the {\rm EOS} of
dense neutron-rich matter. Implications of our findings to neutron star studies
are also discussed.
