The Erdős–Ulam problem on rational distances

Is there a dense set of points in the plane whose pairwise distances are all rational? Expected no, known only conditionally.

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	1	+{
	2	+ "domain": "mathematics",
	3	+ "topics": [
	4	+ "discrete-geometry",
	5	+ "number-theory"
	6	+ ],
	7	+ "type": "proof",
	8	+ "also": [
	9	+ "construction"
	10	+ ],
	11	+ "cover": {
	12	+ "motif": "points",
	13	+ "n": 8
	14	+ },
	15	+ "source": {
	16	+ "key": "erdos",
	17	+ "name": "Erdős Problems",
	18	+ "url": "https://www.erdosproblems.com/212",
	19	+ "id": "212",
	20	+ "curator": "Thomas Bloom",
	21	+ "poser": "Stanisław Ulam",
	22	+ "status": "open",
	23	+ "as_of": "2026-10-20",
	24	+ "catalog": "0521c19bbd38"
	25	+ }
	26	+}