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HadwigerNelson

Hadwiger-Nelson Problem Formalization in Lean 4

Hadwiger-Nelson Problem Formalization in Lean 4

Description

This repository contains a formalized theorem stating that the plane cannot be colored using 4 colors such that no two points at distance 1 from each other have the same color. Specifically, we verify that the 510-vertex graph constructed by Marjin Heule is not 4-colorable.

Installation

  1. Install Lean 4 by following the instructions here.

  2. Install CaDiCaL (SAT solver). You can use a different SAT solver capable of producing LRAT-proofs of unsatisfiability with:

    set_option sat.solver <YOUR_SAT_SOLVER_COMMAND>

  3. Run lake exe cache get to load the Mathlib cache, then build the project with lake build. The building process requires more than 20GB of RAM and takes approximately one hour to complete. Alternatively, you can download oleans from Dropbox, put the content to the .lake directory and only then run lake build.

Proof Overview

We formalize a standard approach to prove the statement:

  1. Parse the .vtx file containing vertices of the non-4-colorable unit distance graph.
  2. Find unit distance edges between its vertices and prove that they have exactly unit length using an automatic tactic.
  3. Reduce the colorability of the finite graph to the satisfiability of some CNF, then use LeanSAT along with an external SAT solver to prove that the graph is non-4-colorable.
  4. Reduce the colorability of the plane to the colorability of finite unit distance graphs and complete the proof.

Building Edges

To find edges of the graph, we cast vertex coordinates to Float and check whether the distance between two given points is close enough to 1. If so, we run the build_edge tactic that combines ring_nf and some extra rewriting of square roots.

Currently, the tactic successfully proves equalities involving numbers that are linear combinations of square roots of naturals over rationals. This is sufficient to build all edges of the 510-vertex Heule graph. Unfortunately, the smallest known non-4-colorable unit distance graph constructed by Jaan Parts is currently beyond the capabilities of build_edge, as it contains numbers like Sqrt[(5*(7 + Sqrt[33]))/2].

TODO

(PRs are welcome)

  • Prove that the 509-vertex Parts graph is a non-4-colorable unit distance graph.
  • Prove that the plane is 7-colorable.
  • Construct the non-4-colorable graph within Lean to make the proof shorter and more independent.

References