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.
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