644.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

2 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 644

References:

erdosproblems.com/644
[EFKT92] Erdős, P. and Fon-Der-Flaass, D. and Kostochka, A. V. and Tuza, Zs., Small transversals in uniform hypergraphs. Siberian Adv. Math. (1992), 82-88.

Imports (1)

FormalConjecturesUtil

theorem Erdos644.erdos_644.parts.i Documented declaration: i sorry line 45 source
Source documentation
Let $f(k,r)$ be minimal such that if $A_1,A_2,\ldots$ is a family of sets, all of size $k$, such that for every collection of $r$ of the $A_is$ there is some pair $\{x,y\}$ which intersects all of the $A_j$, then there is some set of size $f(k,r)$ which intersects all of the sets $A_i$. Is it true that $$f(k,7)=(1+o(1))\frac{3}{4}k?$$ Is it true that for any $r\geq 3$ there exists some constant $c_r$ such that $$f(k,r)=(1+o(1))c_rk?$$

depends on sorry what this means
Statement uses: Iff, True, Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, ENat.toNat, Hypergraph.subfamilyTransversalBound, OfNat.ofNat, HMul.hMul, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_644.parts.i :
    answer(sorry) ↔
    (fun k ↦ ((Hypergraph.subfamilyTransversalBound k 7 2).toNat : ℝ)) ~[atTop]
      (fun k ↦ (3 / 4 : ℝ) * k) := by
…
```
Mentions: Hypergraph.subfamilyTransversalBound
theorem Erdos644.erdos_644.parts.ii For every $r\geq 3$, is there $c_r>0$ such that $f(k,r)=(1+o(1))c_rk$? sorry line 55 source
Source documentation
For every $r\geq 3$, is there $c_r>0$ such that $f(k,r)=(1+o(1))c_rk$?

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Exists, Real, And, LT.lt, Asymptotics.IsEquivalent, Filter.atTop, ENat.toNat, Hypergraph.subfamilyTransversalBound, HMul.hMul
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_644.parts.ii :
    answer(sorry) ↔ ∀ r : ℕ, 3 ≤ r → ∃ c : ℝ, 0 < c ∧
    (fun k ↦ ((Hypergraph.subfamilyTransversalBound k r 2).toNat : ℝ)) ~[atTop]
      (fun k ↦ c * k) := by
…
```
Mentions: c, Hypergraph.subfamilyTransversalBound