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Claim. Theorem 1.3 of R. D. Mauldin, Some problems in set theory, analysis and geometry, in Paul Erdős and his Mathematics I, Bolyai Soc. Math. Stud. 11 (2002), 493–506, which Mauldin states as a result of Chris Freiling and himself: if $A\subseteq\mathbb{R}^2$ contains no three points spanning a triangle of area greater than $1$, then the disk whose area equals the outer measure of $A$ contains no such triangle either, so the outer measure of $A$ is at most $c_0=4\pi/(3\sqrt3)=4\pi/\sqrt{27}$. The proof passes from $A$ to its closed convex hull, which by Lemma 1.4 still contains no triangle of area greater than $1$ and has area at least the outer measure of $A$, and then to Steiner symmetrizations of that convex body, which converge to a disk of the same area. Mauldin presents the theorem as evidence for Erdős's conjecture that $c_0$ is the best constant in Problem 352; the conjecture itself he states as open.

Covers. The convex sets among the measurable $A\subseteq\mathbb{R}^2$ that the question quantifies over, answered yes with any $c>4\pi/\sqrt{27}$: a convex set of measure greater than $4\pi/\sqrt{27}$ contains a triangle of area greater than $1$ by the theorem, and convexity lets that triangle shrink continuously inside the set to one of area exactly $1$. No smaller threshold works, since the open disk of radius $2\cdot3^{-3/4}$ has area $4\pi/\sqrt{27}$ and contains no triangle of area $1$. For nonconvex sets the theorem gives only a triangle of area greater than $1$, which does not answer the question. The convex case also follows from Sas's theorem that every planar convex body contains an inscribed triangle of area at least $3\sqrt3/(4\pi)$ times its own (E. Sas, Über eine Extremumeigenschaft der Ellipsen, Compositio Math. 6 (1939), 468–470), as a comment in the site's thread of 2025-12-27 notes.

Depends on. Nothing in this wiki; the result rests on the cited chapter alone.

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. The author's copy linked above is dated April 23, 2001.

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. The convex case is classical through Sas's refereed theorem, but Sas's paper is not credited by the site and has no page here.