  We examine necessary and sufficient conditions for posterior consistency
under $g$-priors, including extensions to hierarchical and empirical Bayesian
models. The key features of this article are that we allow the number of
regressors to grow at the same rate as the sample size and define posterior
consistency under the sup vector norm instead of the more conventional
Euclidean norm. We consider in particular the empirical Bayesian model of
George and Foster (2000), the hyper-$g$-prior of Liang et al. (2008), and the
prior considered by Zellner and Siow (1980).
