1023.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1023

References:

erdosproblems.com/1023
[Er71, p.105] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.

Imports (1)

FormalConjecturesUtil

def Erdos1023.F F n is the maximal size of a family of subsets of $\{1,\ldots,n\}$ such that no set in this family is the union of other members of the family line 41 source
Source documentation
F n is the maximal size of a family of subsets of $\{1,\ldots,n\}$ such that no set in this family is the union of other members of the family.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.powerset, Finset.Icc, OfNat.ofNat, Finset.SubfamilyUnionFree, Eq, Finset.card
8 instances, coercions and auxiliary constants left out.
```
noncomputable def F (n : ℕ) : ℕ :=
…
```
theorem Erdos1023.erdos_1023 Let $F(n)$ be the maximal size of a family of subsets of $\{1,\ldots,n\}$ such that no set in this family is the union of other members of the family sorry line 54 source
Source documentation
Let $F(n)$ be the maximal size of a family of subsets of $\{1,\ldots,n\}$ such that no set in this family is the union of other members of the family. Is it true that there is a constant $c>0$ such that $$F(n)\sim c \frac{2^n}{n^{1/2}}?$$
Hunter observes in the comments that this follows from the solution to [447], which implies $F(n)\sim \binom{n}{n/2}$.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Asymptotics.IsEquivalent, Nat, Filter.atTop, Erdos1023.F, HDiv.hDiv, HMul.hMul, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1023 : answer(True) ↔
    ∃ c : ℝ, 0 < c ∧
      ((fun n : ℕ => (F n : ℝ)) ~[atTop]
        (fun n : ℕ => c * 2 ^ n / (n : ℝ) ^ (1 / 2 : ℝ))) := by
…
```
Mentions: c, Erdos1023.F
theorem Erdos1023.erdos_1023.variants.erdos_kleitman Erdős and Kleitman proved in unpublished work that $$F(n)\asymp \frac{2^n}{n^{1/2}}.$$ ([Er71] has an exponent of $3/2$, but this is presumably a typo.) sorry line 66 source
Source documentation
Erdős and Kleitman proved in unpublished work that $$F(n)\asymp \frac{2^n}{n^{1/2}}.$$ ([Er71] has an exponent of $3/2$, but this is presumably a typo.)

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos1023.F, HDiv.hDiv, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1023.variants.erdos_kleitman :
    (fun n : ℕ => (F n : ℝ)) =Θ[atTop] (fun n : ℕ => (2 : ℝ) ^ n / (n : ℝ) ^ (1 / 2 : ℝ)) := by
…
```
Mentions: Erdos1023.F
theorem Erdos1023.erdos_1023.variants.hunter Hunter observes in the comments that this follows from the solution to [447], which implies $F(n)\sim \binom{n}{n/2}$ sorry line 75 source
Source documentation
Hunter observes in the comments that this follows from the solution to [447], which implies $F(n)\sim \binom{n}{n/2}$.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos1023.F, Nat.choose, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1023.variants.hunter :
    (fun n : ℕ => (F n : ℝ)) ~[atTop] (fun n : ℕ => (n.choose (n / 2) : ℝ)) := by
…
```
Mentions: Erdos1023.F