  We investigate a possibility of holon pairing for bose condensation based on
the
  Bethe-Salpeter equation obtained from the use of the t-J Hamiltonians of both
t he U(1) and SU(2) slave-boson symmetries. It is shown that the vertex
function contributed from ladder diagram series involving holon-holon
scattering channel in the Bethe-Salpeter equation leads to a singular behavior
at a critical temperature at each hole doping concentration, showing the
instability of the normal state against holon pairing. We find that the holon
pairing instability occurs only in a limited range of hole doping, by showing
an "arch" shaped bose condensation line in agreement with observation for high
$T_c$ cuprates. It is revealed that this is in agreement with a functional
integral approach of the slave-boson theories.
