  We compare state-of-the-art semi-analytic models of galaxy formation as well
as advanced sub-halo abundance matching models with a large sample of
early-type galaxies from SDSS at z < 0.3. We focus our attention on the
dependence of median sizes of central galaxies on host halo mass. The data do
not show any difference in the structural properties of early-type galaxies
with environment, at fixed stellar mass. All hierarchical models considered in
this work instead tend to predict a moderate to strong environmental
dependence, with the median size increasing by a factor of about 1.5-3 when
moving from low to high mass host haloes. At face value the discrepancy with
the data is highly significant, especially at the cluster scale, for haloes
above log Mhalo > 14. The convolution with (correlated) observational errors
reduces some of the tension. Despite the observational uncertainties, the data
tend to disfavour hierarchical models characterized by a relevant contribution
of disc instabilities to the formation of spheroids, strong gas dissipation in
(major) mergers, short dynamical friction timescales, and very short quenching
timescales in infalling satellites. We also discuss a variety of additional
related issues, such as the slope and scatter in the local size-stellar mass
relation, the fraction of gas in local early-type galaxies, and the general
predictions on satellite galaxies.
