  In the paper we probe the possibilities of creating a Kurepa tree in a
generic extension of a model of CH plus no Kurepa trees by an
omega_1-preserving forcing notion of size at most omega_1. In the first section
we show that in the Levy model obtained by collapsing all cardinals between
omega_1 and a strongly inaccessible cardinal by forcing with a countable
support Levy collapsing order many omega_1-preserving forcing notions of size
at most omega_1 including all omega-proper forcing notions and some proper but
not omega-proper forcing notions of size at most omega_1 do not create Kurepa
trees. In the second section we construct a model of CH plus no Kurepa trees,
in which there is an omega-distributive Aronszajn tree such that forcing with
that Aronszajn tree does create a Kurepa tree in the generic extension. At the
end of the paper we ask three questions.
