Triangles of area one in sets of large measure

Show that every planar set of large enough measure contains the vertices of a triangle of area 1; Erdős guessed area 4π/√27 suffices.

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State the problem, from Erdős Problems

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	1	+# Triangles of area one in sets of large measure
	2	+
	3	+If a planar set has enough area, must three of its points span a triangle of area exactly $1$? Erdős showed this for sets of infinite measure, and guessed that area $4\pi/\sqrt{27}\approx2.418$ already suffices, the area of the largest disc that contains no such triangle. Even the existence of some finite threshold is open.
	4	+
	5	+## Problem
	6	+
	7	+Is there a constant $c>0$ such that every Lebesgue measurable set $A\subseteq\mathbb{R}^2$ of measure at least $c$ contains three points that are the vertices of a triangle of area $1$?
	8	+
	9	+Erdős suggested that $c=4\pi/\sqrt{27}$ might work, which would be best possible: an open disc of radius less than $2\cdot3^{-3/4}$ has area close to $4\pi/\sqrt{27}$, and every triangle inscribed in it has area less than $1$.
	10	+
	11	+## Background
	12	+
	13	+Erdős raised the question in the late 1970s in papers on measure-theoretic problems about point sets. He observed that the statement follows from the Lebesgue density theorem when $A$ has infinite measure, or is unbounded with positive measure, so the content of the problem lies in bounded sets of finite measure.
	14	+
	15	+## What is known
	16	+
	17	+- Convex sets. By a theorem of Sas, any convex set of area greater than $4\pi/\sqrt{27}$ contains a triangle of area $1$. Freiling and Mauldin showed that a set of outer measure greater than $4\pi/\sqrt{27}$ contains the vertices of a triangle of area greater than $1$, which settles the problem with the conjectured threshold for compact convex sets.
	18	+- Unions of convex sets. Mauldin (2013) noted that it suffices to treat finite unions of interiors of compact convex sets, and Freiling and Mauldin proved the conjecture for unions of at most three such sets.
	19	+- Sets in a large square. Bulj and Kovač (2026, preprint) showed that a measurable set in $[0,R]^2$ with no triangle of area $1$ has measure $O(R^2/(\log R)^{c})$ for every $c<1/2$, improving an earlier $o(R^2)$ bound of Graham; the conjecture predicts $O(1)$.
	20	+
	21	+## What would count as progress
	22	+
	23	+- A proof that some finite $c$ works, or any bound $O(R^{2-\delta})$ for sets in $[0,R]^2$ without unit-area triangles.
	24	+- The conjecture for unions of four or more convex sets, or for other natural classes such as finite unions of discs of equal radius.
	25	+- A set of measure greater than $4\pi/\sqrt{27}$ with no triangle of area $1$, which would refute Erdős's guess for the constant; for explicit sets this can be checked numerically with rigorous bounds.
	26	+- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/352.lean).
	27	+
	28	+## References
	29	+
	30	+1. P. Erdős, Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange 4 (1978/79), 113–138. [doi:10.2307/44151159](https://doi.org/10.2307/44151159)
	31	+2. R. D. Mauldin, Some problems and ideas of Erdős in analysis and geometry, in Erdős Centennial, Bolyai Soc. Math. Stud. 25 (2013), 365–376. [doi:10.1007/978-3-642-39286-3_13](https://doi.org/10.1007/978-3-642-39286-3_13)
	32	+3. A. Bulj and V. Kovač, On hyperbolic corners and unit-area triangles in planar sets of large measure (2026). [arXiv:2605.30033](https://arxiv.org/abs/2605.30033)
	33	+
	34	+## Source
	35	+
	36	+Imported from [Erdős Problems 352](https://www.erdosproblems.com/352), curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.