FormalConjectures.ErdosProblems.«93»

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import FormalConjecturesUtil

Erdős Problem 93

References:

erdosproblems.com/93

[Er46b] Erdős, P., On sets of distances of {$n$} points. Amer. Math. Monthly (1946), 248--250.

[Er57] Erdős, Paul, Some unsolved problems. Michigan Math. J. (1957), 291-300.

[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.

[Er75f] Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.

[Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.

[Er87b] Erdős, P., Some combinatorial and metric problems in geometry. Intuitive geometry (Siófok, 1985) (1987), 167-177.

[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.

[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.

[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.

[Er97e] Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.

[Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.

[Al63] Altman, E., On a problem of P. Erdős. Amer. Math. Monthly (1963), 148-157.

open EuclideanGeometrynamespace Erdos93

If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor \frac{n}{2}\rfloor$ distinct distances.

Solved by Altman [Al63].

@[category research solved, AMS 52, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos93.lean"]
theorem erdos_93 (A : Finset ℝ²) (hA : ConvexIndep (A : Set ℝ²)) :
    A.card / 2 ≤ distinctDistances A := byA:Finset ℝ²hA:ConvexIndep ↑A⊢ A.card / 2 ≤ distinctDistances A
  sorryAll goals completed! 🐙

end Erdos93