  The problem of constructing confidence sets in the high-dimensional linear
model with $n$ response variables and $p$ parameters, possibly $p\ge n$, is
considered. Full honest adaptive inference is possible if the rate of sparse
estimation does not exceed $n^{-1/4}$, otherwise sparse adaptive confidence
sets exist only over strict subsets of the parameter spaces for which sparse
estimators exist. Necessary and sufficient conditions for the existence of
confidence sets that adapt to a fixed sparsity level of the parameter vector
are given in terms of minimal $\ell^2$-separation conditions on the parameter
space. The design conditions cover common coherence assumptions used in models
for sparsity, including (possibly correlated) sub-Gaussian designs.
