The Number of Atomic Models of Uncountable Theories

We show there exists a complete theory in a language of size continuum possessing a unique atomic model which is not constructible. We also show it is consistent with $ZFC+{\mathrm{\aleph }}_1<2^{{\mathrm{\aleph }}_0}$ that there is a complete theory in a language of size ${\mathrm{\aleph }}_1$ possessing a unique atomic model which is not constructible. Finally we show it is consistent with $ZFC + \aleph_1 < 2^{\aleph_0}$thatforeverycompletetheory$T$inalanguageofsize$\aleph_1$,if$T$ has uncountable atomic models but no constructible models, then $T$ has $2^{{\mathrm{\aleph }}_1}$ atomic models of size ${\mathrm{\aleph }}_1$.