130.lean`

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L4DEL: Lean4 Domain Explanation Language

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 130

Reference: erdosproblems.com/130

Imports (1)

FormalConjecturesUtil

theorem Erdos130.erdos_130 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 sorry line 44 source
Source documentation
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.

depends on sorry what this means
Statement uses: Iff, True, Exists, Set, EuclideanSpace, Real, Fin, OfNat.ofNat, Nat, And, Set.Infinite, EuclideanGeometry.InGeneralPosition, Eq, ENat, SimpleGraph.chromaticNumber, SimpleGraph.IntegerDistancePlaneGraph, Top.top
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_130 :
    answer(True) ↔
      ∃ A : Set ℝ², A.Infinite ∧ InGeneralPosition A ∧
        (IntegerDistancePlaneGraph A).chromaticNumber = ⊤ := by
…
```