  The Falicov-Kimball model has been known to phase separate when the
correlation strength is large and the temperature is low. We examine the
stability of phase separation under the influence of an additional
electron-phonon interaction that tries to force the electron density to be
homogeneous. We find that the electron-phonon interaction eventually destroys
the phase-separated state once it becomes larger than some critical value. The
results of this work may have an influence on theories of collossal
magentoresistance materials.
