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Claim. In P. Erdős, Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978/79), no. 2, 113–138, p. 122, Erdős states that if $S\subseteq\mathbb{R}^2$ has infinite planar measure, then for every $a>0$ the set $S$ contains three points spanning a triangle of area $a$; the proof, an easy consequence of the Lebesgue density theorem, is left as an exercise. He adds that he had published the statement as a problem in Matematikai Lapok and that a reader proved a slightly stronger theorem, that it suffices for some line to meet $S$ in a set of positive linear measure and for $S$ to have points arbitrarily far from that line; that the triangle may be taken isosceles or right-angled; and that some set of infinite measure contains no equilateral triangle of area $1$. The question that is Problem 352 follows on the same page. Mauldin's 2002 chapter (Freiling and Mauldin 2002) records that Erdős would sketch the argument through Steinhaus's theorem on difference sets and states, without a printed proof, that an unbounded set of positive measure also contains the vertices of a triangle of area $1$; the site's commentary credits both statements to Erdős as unpublished. The source card is Erdős 1978.

Covers. The measurable $A\subseteq\mathbb{R}^2$ of infinite measure, answered yes for every $c$, and, on the statement of Mauldin's chapter and the site's commentary, the unbounded sets of positive measure. Bounded sets of finite measure, where the question lives, are untouched.

Depends on. Nothing in this wiki; the result rests on the cited survey and the sources that repeat it.

Dating. The page is dated by the volume's year; the survey appeared in volume 4, number 2, of Real Anal. Exchange, which the record gives as 1978/79, and the day in the page name is a placeholder.

Acceptance. None listed. No printed proof is cited: Erdős left the argument to the reader, and the site's curator records the result as unpublished while labeling the problem OPEN, which is not acceptance of a claim. The statement is reproduced as known in Mauldin's chapters of 2002 and 2013.