83.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, 3 contain sorry, 0 declared as axioms

Erdős Problem 83

References:

erdosproblems.com/83
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[Er92e] Erdős, Pál, Some Unsolved problems in Geometry, Number Theory and Combinatorics. Eureka (1992), 44-48.
[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
[ErKoRa61] Erdős, P. and Ko, Chao and Rado, R., Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2) (1961), 313-320.
[AhKh97] Ahlswede, Rudolf and Khachatrian, Levon H., The complete intersection theorem for systems of finite sets. European J. Combin. (1997), 125-136.

Imports (1)

FormalConjecturesUtil

theorem Erdos83.erdos_83 sorry line 64 source
depends on sorry what this means
Statement uses: Nat, Finset, Fin, HMul.hMul, OfNat.ofNat, Membership.mem, Eq, Finset.card, LE.le, Inter.inter, HDiv.hDiv, HSub.hSub, Nat.choose, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos83.erdos_83.variants.extremal The bound in erdős_83 is attained by the family of all subsets of $[4n]$ of size $2n$ containing at least $n+1$ elements from $[2n]$ sorry line 74 source
Source documentation
The bound in erdős_83 is attained by the family of all subsets of $[4n]$ of size $2n$ containing at least $n+1$ elements from $[2n]$.

L4DEL explanation
The bound in erdős_83 is attained by the family of all subsets of $[4n]$ of size $2n$ containing at least $n+1$ elements from $[2n]$.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Nat, Exists, Finset, Fin, HMul.hMul, OfNat.ofNat, And, Membership.mem, Eq, Finset.card, LE.le, Inter.inter, HDiv.hDiv, HSub.hSub, Nat.choose, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
32 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_83.variants.extremal (n : ℕ) :
    ∃ F : Finset (Finset (Fin (4 * n))), (∀ A ∈ F, A.card = 2 * n) ∧
      (∀ A ∈ F, ∀ B ∈ F, 2 ≤ (A ∩ B).card) ∧
      F.card = ((4 * n).choose (2 * n) - (2 * n).choose n ^ 2) / 2 := by
…
```
theorem Erdos83.erdos_83.variants.ahlswede_khachatrian Documented declaration: ahlswede_khachatrian sorry line 87 source
Source documentation
The complete intersection theorem of Ahlswede and Khachatrian [AhKh97]: for $2\leq t\leq k\leq m$, every family of $k$-subsets of $[m]$ with pairwise intersections of size at least $t$ is at most as large as one of the families of all $k$-subsets of $[m]$ containing at least $t+r$ elements of $\{1,\ldots,t+2r\}$, $r \geq 0$.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Finset, Fin, Membership.mem, Eq, Finset.card, Inter.inter, Exists, Finset.filter, And, HAdd.hAdd, LT.lt, HMul.hMul, Finset.univ
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_83.variants.ahlswede_khachatrian (m k t : ℕ) (ht : 2 ≤ t) (htk : t ≤ k)
    (hkm : k ≤ m) (F : Finset (Finset (Fin m))) (hF : ∀ A ∈ F, A.card = k)
    (hinter : ∀ A ∈ F, ∀ B ∈ F, t ≤ (A ∩ B).card) :
    ∃ r : ℕ, F.card ≤ ({A : Finset (Fin m) | A.card = k ∧
      t + r ≤ (A.filter fun i : Fin m => (i : ℕ) < t + 2 * r).card} : Finset _).card := by
…
```