{"id":"palomar/PALOMAR-2026-09-02-000005","url":"https://github.com/sneed-and-feed/lean-theorems-1/tree/afbbe0c3c1823bb84502f83f05936d1836f8cc24","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-09-02-000005-v1.json","repo":"https://github.com/sneed-and-feed/lean-theorems-1","tags":["math.CO","math.MG","msc:05B25","msc:51E10","msc:51A15","incidence-geometry","combinatorics","lean","linear-spaces"],"score":0,"title":"The De Bruijn–Erdős Theorem on Incidence Geometry and Near-Pencil Tightness","trust":"high","domain":"math.CO","source":"palomar","authors":["Noah Cohen"],"reviews":0,"subject":"math.CO","summary":"A complete machine-checked Lean 4 formalization of the De Bruijn–Erdős theorem (1948) in incidence geometry and pairwise balanced block designs, together with its constructive near-pencil tightness proof. Proves that any finite linear space on n ≥ 3 non-collinear points determines at least n lines (|P| ≤ |L|) via the classical combinatorial double-counting inequality. Furthermore, explicitly constructs the near-pencil configuration with designated apex p₀ (consisting of one line of n - 1 points and n - 1 lines of 2 points) and proves that it forms a valid linear space achieving exact…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-09-02","reactions":{},"formalized":true,"formalizes":[{"title":"On a combinatorial problem","identifier":"","relationship":"formalizes"}]}