  We present the results of a survey for Extremely Red Objects (EROs) in the
fields of ten galaxy cluster lenses at z~0.2, combining high-resolution Hubble
Space Telescope imaging with deep K-band imaging from UKIRT. We detect 60 EROs
with R-K>=5.3, of which 26 have R-K>=6.0 in a total image plane survey area of
49arcmin^2 down to K=20.6. We use detailed models of the cluster lenses to
quantify the lens amplification and thus correct the observed number counts and
survey area for the effects of gravitational lensing. After making these
corrections, we estimate surface densities at K<=21.6 of 2.5+/-0.4 and
1.2+/-0.3 arcmin^-2 for EROs with R-K>=5.3 and 6.0 respectively. These ERO
number counts agree with previous shallower surveys at K<~19 and flatten
significantly at magnitudes fainter than K~19-20. This flattening may be due to
a transition from an ERO population dominated by evolved galaxies at z~1-2
(K<~19.5) to one dominated by dusty starburst galaxies at z>1 (K>~19.5). We
also compare our results with various model predictions, including a model that
attempts to explain EROs in terms of a single population of elliptical galaxies
formed at high redshift. We find that a formation epoch of z_f~2.5 for this
population matches the observed surface density of R-K>=5.3 EROs quite well,
and the R-K>=6.0 sample less well. More sophisticated models, including
semi-analytic prescriptions, under-predict the ERO surface density by
approximately an order of magnitude, suggesting that these models produce
insufficient stars and/or dust at high redshift.
