  The sensitive correlations between the low-density halo structure and the
high-density properties of the nuclear equation of state (EOS) are constructed
in light kaonic nuclei with the relativistic mean-field theory. More
specifically, the $1p_{1/2}$ halo spreads out linearly with increasing the
pressure and sound velocity square at supra-normal densities and quadratically
with decreasing the incompressibility at saturation density. These results
suggest that the novel halo in light kaonic nuclei can serve as a sensitive
indicator of the nuclear EOS of symmetric matter at supra-normal densities.
