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	"review": true,
27	"approval": "first-time"
28	}