In the paper we formulate a Covering Property Axiom CPA_{cube}^{game},
which holds in the iterated perfect set model,
and show that it implies
the existence of uncountable strong \gamma-sets in **R**
(which are strongly meager)
as well as uncountable \gamma-sets in **R**
are not strongly meager. These sets must be
of cardinality \omega_{1}, since
every \gamma-set is universally null, while
CPA_{cube}^{game} implies that every universally null
has cardinality less than \continuum=\omega_{2}.

We will also show that CPA_{cube}^{game} implies the existence of
a partition of **R** into \omega_{1} null compact sets.

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**Last modified June 10, 2003.**