Resolving Erdős–Ulam Monochromatic Union-Closed Family Conjectures

We prove both Erdős–Ulam conjectures on monochromatic union-closed families (Erdős Problem 1183). Every two-colouring of the subsets of a finite set contains a monochromatic union-closed family whose size grows faster than any fixed power of the size of the set, while suitable colourings admit no monochromatic union-closed family of exponential size. Both results hold for any number of colours and are verified in Lean.

This record contains the paper (PDF and LaTeX source with TikZ figures) and the Lean 4 formalisation with Mathlib. Every theorem, proposition, corollary and lemma of the paper is checked in Lean with no sorry, and each audited statement uses only the standard axioms propext, Classical.choice and Quot.sound. Claude (Anthropic) assisted with the coding.