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Claim. Section 5 of R. D. Mauldin, Some problems and ideas of Erdős in analysis and geometry, in Erdős Centennial, Bolyai Soc. Math. Stud. 25 (2013), 365–376, which the site credits to Freiling and Mauldin. Mauldin first restates Problem 352 in an equivalent form (Mauldin's Problem 5.2, obtained by standard approximations in measure theory that Mauldin does not spell out): is there a finite constant $C$ such that every set $E$ that is the union of the interiors of at most $n$ compact convex sets and has measure greater than $C$ contains the vertices of a triangle of area $1$, and is the best constant $c_0=4\pi/(3\sqrt3)=4\pi/\sqrt{27}$? Mauldin then shows that $c_0$ is the best possible constant for $n\le3$: the cases $n=1$ and $n=2$ come from Mauldin's 2002 chapter (a convex body, or the convex hull of two such bodies, containing no triangle of area greater than $1$ has area at most $c_0$), and for $n=3$ the author writes out a redistribution-of-mass argument in which a small triple reduces to $n=1$ through its convex hull, while for a large triple the region swept out between the two larger bodies has area at least that of the smallest, so the three bodies can be replaced by one. The general case Mauldin leaves open. The source card is Mauldin 2013.

Covers. The sets $A$ that are the union of the interiors of at most three compact convex sets, answered yes with any $c>4\pi/\sqrt{27}$ in the form Mauldin's Problem 5.2 states, the open disk of area $4\pi/\sqrt{27}$ showing that no smaller threshold works. The equivalence of Problem 5.2 with the question for all measurable sets needs every finite $n$, so the general question is untouched.

Depends on. Freiling and Mauldin 2002 for the cases $n=1$ and $n=2$.

Dating. The page is dated by the volume's year; the chapter record gives no month, and the day in the page name is a placeholder.

Acceptance. None listed. The chapter appeared in a Bolyai Society volume, not a journal, and no evidence that the volume was refereed is recorded. The site's curator credits the result in the problem's commentary while labeling the problem OPEN, which is not acceptance of a claim.