Red-sequence galaxies record the history of terminated star-formation in the
Universe and can thus provide important clues to the mechanisms responsible for
this termination. We construct composite samples of published cluster and field
galaxy photometry in order to study the build-up of galaxies on the
red-sequence, as parameterised by the dwarf-to-giant ratio (DGR). We find that
the DGR in clusters is higher than that of the field at all redshifts, implying
that the faint end of the red-sequence was established first in clusters. We
find that the DGR evolves with redshift for both samples, consistent with the
``down-sizing'' picture of star formation. We examine the predictions of
semi-analytic models for the DGR and find that neither the magnitude of its
environmental dependence nor its evolution is correctly predicted in the
models. Red-sequence DGRs are consistently too high in the models, the most
likely explanation being that the strangulation mechanism used to remove hot
gas from satellite galaxies is too efficient. Finally we present a simple toy
model including a threshold mass, below which galaxies are not strangled, and
show that this can predict the observed evolution of the field DGR.Comment: MNRAS letters accepted. 5 pages, 1 figur