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Suppose that $X$ is a Banach space. We will show that $X$ does not contain acopy of $c_0$ if and only if for each series which is not unconditionallyconvergent in $X$ respective sets coding all bounded subseries andrearrangements are meager. We use Bessaga-Pe{\l}czy{\'n}ski $c_0-$Theorem andconcept of uniformly unconditionally bounded series. Moreover we prove similarresult for the ideal boundedness for a class of Baire ideals using Talagrand'scharacterisation.

Ideal boundedness of subseries and rearrangements in Banach spaces vs Banach spaces possessing a copy of $c_0$