{"id":"hexagon/2611.00070","pdf":"https://hexagonmath.org/pdf/2611.00070v1","url":"https://hexagonmath.org/abs/2611.00070v1","tags":["cs.DM","cs.LO","msc:05D99","msc:00A08","msc:68V20","Hall's theorem","Lean formalisation","MacMahon cubes","colour cubes","extremal combinatorics"],"score":0,"title":"Three-colour solutions from four-colour cubes","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Jason Davies"],"reviews":0,"subject":"math.CO","summary":"For every $n\\ge2$, any $16n-3$ cubes, each using all four colours of a fixed palette, contain an $n\\times n\\times n$ frame whose exterior faces are monochromatic in exactly three colours; this bound is sharp. For a fixed palette, exactly two collections of $16n-4$ cubes fail, and reflection exchanges them. For $n\\ge3$, every failure of size at least $14n$ lies in one ordinary orientation class with four bounded colour buckets. This gives $2\\binom{d+15}{15}$ failures at size $16n-4-d$, for $0\\le d\\le2n-4$, of which $2\\binom{d+11}{11}$ are inclusion-maximal. The threshold for a complete cube…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Code","Codex"],"discussed":0,"published":"2026-11-04","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}