780.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 780

References:

erdosproblems.com/780
[Er76] Erdős, Paul, Problems and results in combinatorial analysis. Colloquio Internazionale sulle Teorie Combinatorie (Roma, 1973), Tomo II (1976), 3-17.
[Lo78] Lovász, L., Kneser's conjecture, chromatic number, and homotopy. J. Combin. Theory Ser. A (1978), 319-324.
[AFL86] Alon, N. and Frankl, P. and Lovász, L., The chromatic number of Kneser hypergraphs. Trans. Amer. Math. Soc. (1986), 359-370.

Imports (1)

FormalConjecturesUtil

theorem Erdos780.erdos_780 sorry line 47 source
depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, HAdd.hAdd, HMul.hMul, HSub.hSub, Subtype, Finset, Fin, Eq, Finset.card, Exists, And, Pairwise, Disjoint
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 Erdos780.erdos_780.variants.sharp Documented declaration: sharp sorry line 60 source
Source documentation
This would be best possible: if $n=kr-1+(t-1)(k-1)$ then decomposing $[n]$ as one set $X_1$ of size $kr-1$ and $t-1$ sets $X_2,\ldots,X_{t}$ of size $k-1$, a colouring without $k$ pairwise disjoint edges is given colouring all subsets of $X_1$ in colour $1$ and assigning an edge with colour $2\leq i\leq t$ if $i$ is minimal such that $X_i$ intersects the edge.

depends on sorry what this means
Statement uses: Nat, LE.le, OfNat.ofNat, Exists, Subtype, Finset, Fin, HAdd.hAdd, HSub.hSub, HMul.hMul, Eq, Finset.card, Not, And, Pairwise, Disjoint
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_780.variants.sharp (k r t : ℕ) (hk : 1 ≤ k) (hr : 1 ≤ r) (ht : 1 ≤ t) :
    ∃ c : {e : Finset (Fin (k * r - 1 + (t - 1) * (k - 1))) // e.card = r} → Fin t,
      ¬ ∃ (i : Fin t) (e : Fin k → {e : Finset (Fin (k * r - 1 + (t - 1) * (k - 1))) //
          e.card = r}), (∀ j, c (e j) = i) ∧ Pairwise fun a b ↦ Disjoint (e a).1 (e b).1 := by
…
```
Mentions: c