On perfectly meager sets (Q2781270)

A subset \(X_0\) of a Polish space \(X\) is perfectly meager (PM) if for every perfect set \(P\subset X\), \(X_0\cap P\) is meager in \(P\) with the relative topology. \(X_0\) is universally meager (UM) if every Borel isomorphic image of \(X_0\) in \(X\) is meager. It is clear that \(\text{UM}\subset\text{PM}\), but the other inclusion my fail. In fact, \textit{W. Sierpiński} proved that CH implies \(\text{PM}\neq\text{UM}\) [Fundam. Math. 22, 262-266 (1934; Zbl 0009.13202)]. In 1935 Marczewski asked whether the product of perfectly meager sets is perfectly meager. Under CH this question has been answered in the negative by \textit{I. Recław} [Proc. Am. Math. Soc. 112, No. 4, 1029-1031 (1991; Zbl 0734.04002)]. Unlike PM, the class of UM sets is closed under products [\textit{P. Zakrzewski}, Proc. Am. Math. Soc. 129, No. 6, 1793-1798 (2001; Zbl 0967.03043)]. In this paper, the author shows that it is consistent with ZFC that \(\text{PM}=\text{UM}\). In particular, it is also consistent with ZFC that the product of two PM sets is PM. Thus, Marczewski's question is undecidable in ZFC.