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 \omega₁, since every \gamma-set is universally null, while CPA_cube^game implies that every universally null has cardinality less than \continuum=\omega₂.

We will also show that CPA_cube^game implies the existence of a partition of R into \omega₁ null compact sets.

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Last modified June 10, 2003.