502.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 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.

Imports (1)

FormalConjecturesUtil

theorem Erdos502.erdos_502 What is the size of the largest $A\subseteq \mathbb{R}^n$ such that there are only two distinct distances between elements of $A$? sorry line 52 source
Source documentation
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]).

depends on sorry what this means
Statement uses: Nat, Set, EuclideanSpace, Real, Fin, Set.Finite, Eq, Set.ncard, Set.ofPred, Exists, And, Membership.mem, Ne, Dist.dist, OfNat.ofNat, ENNReal, LE.le, Nat.choose, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
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 := by
…
```