>**Problem.** Let $X$ be a Polish space, $\mathcal I$ be a $\sigma$-ideal with Borel base, and $\mathcal A\subset\mathcal I$ be a point-finite cover of $X$. Is it true that $\mathcal A$ conatins a subfmaily $\mathcal B$ whose union $\bigcup\mathcal B$ is *completely $\mathcal I$-nonmeasurable* in the sense that any Borel subset $B\notin\mathcal I$ intersects both sets $\bigcup\mathcal B$ and $X\setminus\bigcup\mathcal B$.  

(This problem (attributed to Jacek Cichoń) was written 27.07.2018 by Szymon Żeberski from Wrocław on [page 28][1] of [Volume 2][2] of the [Lviv Scottish Book][3]).


  [1]: http://www.math.lviv.ua/szkocka/viewpage.php?vol=2&page=28
  [2]: http://www.math.lviv.ua/szkocka/viewbook.php?vol=2
  [3]: http://www.math.lviv.ua/szkocka