650.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 650

References:

erdosproblems.com/650
[Er78] Erdős, Paul, Problems and results in combinatorial analysis and combinatorial number theory. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40.
[Er86c] Erdős, P., Some problems on number theory. (1986), 53-67.
[Er95c] Erdős, Paul, Some problems in number theory. Octogon Math. Mag. (1995), 3-5.
[ErSu59] Erdős, Pál and Surányi, János, Bemerkungen zu einer Aufgabe eines mathematischen Wettbewerbs. Mat. Lapok (1959), 39-48.
[VLT26] W. Van Doorn, Y. Li, and Q. Tang, Optimal bounds for an Erdős problem on matching integers to distinct multiples. arXiv:2603.28636 (2026).

Imports (1)

FormalConjecturesUtil

def Erdos650.f Documented declaration: f line 44 source
Source documentation
f m is the largest r such that whenever A ⊆ {1, …, N} has m elements, every interval in [1, ∞) of length 2 * N contains r distinct integers b 0, …, b (r - 1), each b i being divisible by some a i ∈ A, where a 0, …, a (r - 1) are distinct.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Finset, LE.le, Finset.Icc, OfNat.ofNat, Eq, Finset.card, Real, Exists, Fin, And, Function.Injective, Membership.mem, Set, Set.Ioo, HAdd.hAdd, HMul.hMul, Dvd.dvd
23 instances, coercions and auxiliary constants left out.
```
noncomputable def f (m : ℕ) : ℕ :=
…
```
theorem Erdos650.erdos_650.parts.i sorry line 66 source
depends on sorry what this means
Statement uses: Nat, Eq, Erdos650.f, Min.min, Nat.ceil, Real, HMul.hMul, OfNat.ofNat, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos650.erdos_650.parts.ii sorry line 86 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, Real, Erdos650.f, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos650.erdos_650.variants.erdos_suranyi Erdős and Surányi [ErSu59] proved that $f(m)\geq\sqrt{m}$ sorry line 93 source
Source documentation
Erdős and Surányi [ErSu59] proved that $f(m)\geq\sqrt{m}$.

depends on sorry what this means
Statement uses: Nat, LE.le, Real, Real.sqrt, Erdos650.f
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_650.variants.erdos_suranyi (m : ℕ) : Real.sqrt m ≤ (f m : ℝ) := by
…
```
Mentions: Erdos650.f
theorem Erdos650.erdos_650.variants.erdos_selfridge Erdős and Selfridge proved (see [Er78] and [Er86c]) that $f(m^2)\leq 2m$, which implies $f(m)\leq 2\lceil \sqrt{m}\rceil$ for all $m$ sorry line 101 source
Source documentation
Erdős and Selfridge proved (see [Er78] and [Er86c]) that $f(m^2)\leq 2m$, which implies $f(m)\leq 2\lceil \sqrt{m}\rceil$ for all $m$.

depends on sorry what this means
Statement uses: Nat, LE.le, Erdos650.f, HPow.hPow, OfNat.ofNat, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_650.variants.erdos_selfridge (m : ℕ) : f (m ^ 2) ≤ 2 * m := by
…
```
Mentions: Erdos650.f