447.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

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 447

References:

erdosproblems.com/447
[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
[Kl71] Kleitman, Daniel, Collections of subsets containing no two sets and their union. Proceedings of the LA Meeting AMS (1971), 153-155.

Imports (1)

FormalConjecturesUtil

def Erdos447.maxUnionFree The largest size of a union-free collection $\mathcal{F}$ of subsets of $[n]$ line 38 source
Source documentation
The largest size of a union-free collection $\mathcal{F}$ of subsets of $[n]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, Fin, And, Finset.UnionFree, Eq, Finset.card
2 instances, coercions and auxiliary constants left out.
```
noncomputable def maxUnionFree (n : ℕ) : ℕ :=
…
```
theorem Erdos447.erdos_447.parts.i How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? sorry line 50 source
Source documentation
How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean there are no solutions to $A\cup B=C$ with distinct $A,B,C\in \mathcal{F}$. Must $\lvert \mathcal{F}\rvert =o(2^n)$?
In [Er65b] Erdős reported that the estimate $\lvert \mathcal{F}\rvert=o(2^n)$ was proved in unpublished work by Sárközy and Szemerédi.

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsLittleO, Nat, Real, Filter.atTop, Erdos447.maxUnionFree, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_447.parts.i : answer(True) ↔
    (fun n : ℕ => (maxUnionFree n : ℝ)) =o[atTop] (fun n : ℕ => (2 : ℝ) ^ n) := by
…
```
Mentions: Erdos447.maxUnionFree
theorem Erdos447.erdos_447.parts.ii How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? sorry line 63 source
Source documentation
How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be? By union-free we mean there are no solutions to $A\cup B=C$ with distinct $A,B,C\in \mathcal{F}$. Perhaps even $$\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}?$$
Solved by Kleitman [Kl71], who proved $$\lvert \mathcal{F}\rvert <(1+o(1))\binom{n}{\lfloor n/2\rfloor}.$$

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Asymptotics.IsLittleO, Filter.atTop, OfNat.ofNat, Filter.Eventually, LT.lt, Erdos447.maxUnionFree, HMul.hMul, HAdd.hAdd, Nat.choose, HDiv.hDiv
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 erdos_447.parts.ii : answer(True) ↔
    ∃ c : ℕ → ℝ, (c =o[atTop] (1 : ℕ → ℝ)) ∧ ∀ᶠ n : ℕ in atTop,
      (maxUnionFree n : ℝ) < (1 + c n) * (n.choose (n / 2) : ℝ) := by
…
```
Mentions: c, Erdos447.maxUnionFree