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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	1	+{
	2	+ "domain": "mathematics",
	3	+ "topics": [
	4	+ "analysis",
	5	+ "discrete-geometry"
	6	+ ],
	7	+ "type": "proof",
	8	+ "source": {
	9	+ "key": "erdos",
	10	+ "name": "Erdős Problems",
	11	+ "url": "https://www.erdosproblems.com/352",
	12	+ "id": "352",
	13	+ "curator": "Thomas Bloom",
	14	+ "poser": "Paul Erdős",
	15	+ "status": "open",
	16	+ "as_of": "2026-10-20",
	17	+ "catalog": "0521c19bbd38"
	18	+ }
	19	+}