FormalConjectures.ErdosProblems.«502»

/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

Erdős Problem 502

References:

erdosproblems.com/502

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

[BBS83] Bannai, Eiichi and Bannai, Etsuko and Stanton, Dennis, An upper bound for the cardinality of an $s$-distance subset in real Euclidean space. II. Combinatorica (1983), 147-152.

[PePo21] Petrov, Fedor and Pohoata, Cosmin, A remark on sets with few distances in $\mathbb{R}^d$. Proc. Amer. Math. Soc. (2021), 569-571.

open scoped EuclideanGeometrynamespace Erdos502

What is the size of the largest $A\subseteq \mathbb{R}^n$ such that there are only two distinct distances between elements of $A$? That is, $$# { \lvert x-y\rvert : x\neq y\in A} = 2.$$

Asked to Erdős by Coxeter. Bannai, Bannai, and Stanton [BBS83] have proved that $$\lvert A\rvert \leq \binom{n+2}{2}.$$ A simple proof of this upper bound was given by Petrov and Pohoata [PePo21].

The exact maximum is not known in general: a lower bound of $\binom{n+1}{2}$ follows from the construction of Alweiss (see [503]).

@[category research solved, AMS 51 52, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos502.lean"]
theorem erdos_502 (n : ℕ) (A : Set (ℝ^n)) (hA : A.Finite)
    (hA2 : {d : ℝ | ∃ x ∈ A, ∃ y ∈ A, x ≠ y ∧ dist x y = d}.ncard = 2) :
    A.ncard ≤ (n + 2).choose 2 := byn:ℕA:Set (ℝ^n)hA:A.FinitehA2:{d | ∃ x ∈ A, ∃ y ∈ A, x ≠ y ∧ dist x y = d}.ncard = 2⊢ A.ncard ≤ (n + 2).choose 2
  sorryAll goals completed! 🐙

end Erdos502