{"id":"palomar/PALOMAR-2026-08-19-000002","url":"https://github.com/elliotglazer/erdos501/tree/218d1c1e46f77d4db80e566d1721782e85b94a17","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-08-19-000002-v3.json","repo":"https://github.com/elliotglazer/erdos501","tags":["math.LO","math.CO","msc:03E35","msc:03E75","msc:03E40","msc:03E50","msc:05D10","msc:28A05","set-theory","independence"],"score":0,"title":"elliotglazer/erdos501","trust":"high","domain":"math.LO","source":"palomar","authors":["Elliot Glazer"],"reviews":0,"subject":"math.LO","summary":"Erdős problem #501 (Erdős 1961; Erdős–Hajnal 1971, Problem 38): for every real x let A_x be a bounded set of reals of Lebesgue outer measure < 1; must there be an infinite independent set, i.e. an infinite X ⊆ ℝ with x ∉ A_y for all distinct x, y ∈ X? And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-06","reactions":{},"formalized":true,"formalizes":[{"title":"Infinite free set for small measure set mappings","identifier":"Proc. Amer. Math. Soc. 100 (1987), 335–339","relationship":"formalizes"},{"title":"Directed graphs over topological spaces: some set theoretical aspects","identifier":"Israel J. Math. 11 (1972), 231–248","relationship":"formalizes"},{"title":"Erdős Problem 501 after adding ω₂ random reals (draft, revision 10)","identifier":"docs/paper/erdos501_random_profiles_rev10.pdf (unpublished draft, 2026-08-16)","relationship":"adapts"}]}