FormalConjectures.ErdosProblems.«130»

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import FormalConjecturesUtil

Erdős Problem 130

Reference: erdosproblems.com/130

namespace Erdos130open EuclideanGeometry SimpleGraph

Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle. Consider the graph with vertices the points in $A$, where two vertices are joined by an edge if and only if they are an integer distance apart. How large can the chromatic number and clique number of this graph be? In particular, can the chromatic number be infinite?

The chromatic number can be infinite: there is an infinite general-position set whose integer-distance graph admits no finite proper colouring. How large the clique number can be is not addressed here.

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/williamjblair/lean-proofs/blob/4f915a323443bfb1709a6805a013812016dca88a/starfleet/erdos-130/Research/Basic.lean"]
theorem erdos_130 :
    answer(True) ↔
      ∃ A : Set ℝ², A.Infinite ∧ InGeneralPosition A ∧
        (IntegerDistancePlaneGraph A).chromaticNumber = ⊤ := by⊢ True ↔ ∃ A, A.Infinite ∧ InGeneralPosition A ∧ χ(IntegerDistancePlaneGraph A) = ⊤
  sorryAll goals completed! 🐙

end Erdos130