{"id":"palomar/PALOMAR-2026-11-03-000006","url":"https://github.com/ukniusanqian-alt/erdos654-lean/tree/3146ec1b4c120e366c10bbb92dec8de596cd7aa1","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-03-000006-v1.json","repo":"https://github.com/ukniusanqian-alt/erdos654-lean","tags":["math.CO","msc:05C35"],"score":0,"title":"Erdős Problem 1033: a butterfly blow-up beating 2(√3 - 1)","trust":"high","domain":"math.CO","source":"palomar","authors":["ukniusanqian-alt"],"reviews":0,"subject":"math.CO","summary":"Lean 4 / Mathlib proof that the Bollobás–Erdős question of Erdős Problem 1033 has a negative answer. For every t ≥ 1 an explicit blow-up of the butterfly graph (two triangles sharing a vertex) on n = 668t vertices has 111561t² > n²/4 edges, and every triangle has degree sum exactly 978t, while 978/668 < 2(√3 - 1). Hence it is false that every graph on n vertices with more than n²/4 edges has a triangle with degree sum at least (2(√3 - 1) - o(1))n.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-03","reactions":{},"formalized":true,"formalizes":[{"title":"Optimized density argument for Erdős 1033","identifier":"https://woutercvb.github.io/Erdos1033_smalloptimization.pdf","relationship":"adapts"}]}