Problem #352: Is there some $c>0$ such that every measurable $A\subseteq...

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Progress

Erdős (unpublished) proved that this is true if $A$ has infinite measure, or if $A$ is an unbounded set of positive measure (stating in [Er78d] and [Er83d] it 'follows easily from the Lebesgue density theorem').

In [Er78d] and [Er83d] he speculated that perhaps $C=4\pi/\sqrt{27}\approx 2.418$ works, which would be the best possible, as witnessed by a circle of radius $<2\cdot 3^{-3/4}$.

Further evidence for this is given by a result of Freiling and Mauldin [Ma02], who proved that if $A$ has outer measure $>4\pi/\sqrt{27}$ then $A$ contains the vertices of a triangle with area $>1$. This also proves the same threshold for the original problem under the assumption that $A$ is a compact convex set.

Mauldin also discusses this problem in [Ma13], in which he notes that it suffices to prove this under the assumption that $A$ is the union of the interiors of $n<\infty$ many compact convex sets. Freiling and Mauldin (see [Ma13]) have proved this conjecture if $1\leq n\leq 3$.