1025.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, 1 contain sorry, 0 declared as axioms

Erdős Problem 1025

References:

erdosproblems.com/1025
[Er71] Erdős, P., Some unsolved problems in graph theory and combinatorial analysis. Combinatorial Mathematics and its Applications (Proc. Conf., Oxford, 1969) (1971), 97-109.
[ErHa58] Erdős, P. and Hajnal, A., On the structure of set mappings. Acta Math. Acad. Sci. Hungar. (1958), 111-133.
[Sp72] Spencer, Joel, Turán's theorem for $k$-graphs. Discrete Math. (1972), 183--186.
[CFS16] Conlon, David and Fox, Jacob and Sudakov, Benny, Short proofs of some extremal results II. J. Combin. Theory Ser. B (2016), 173--196.

Imports (1)

FormalConjecturesUtil

def Erdos1025.IsIndependent X is independent for a set mapping f on pairs if f s(x, y) ∉ X whenever x ≠ y are in X line 42 source
Source documentation
X is independent for a set mapping f on pairs if f s(x, y) ∉ X whenever x ≠ y are in X.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Sym2, Fin, Finset
Proof uses: Membership.mem, Ne, Not, Sym2.mk
2 instances, coercions and auxiliary constants left out.
```
def IsIndependent {n : ℕ} (f : Sym2 (Fin n) → Fin n) (X : Finset (Fin n)) : Prop :=
…
```
def Erdos1025.g g n is the largest k such that every set mapping f on the pairs of an n-element set with f(x, y) ∉ {x, y} admits an independent set of size at least k line 47 source
Source documentation
g n is the largest k such that every set mapping f on the pairs of an n-element set with f(x, y) ∉ {x, y} admits an independent set of size at least k.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, And, LE.le, Sym2, Fin, Ne, Sym2.mk, Exists, Finset, Erdos1025.IsIndependent, Finset.card
2 instances, coercions and auxiliary constants left out.
```
noncomputable def g (n : ℕ) : ℕ :=
…
```
theorem Erdos1025.erdos_1025 sorry line 66 source
depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos1025.g, Real.sqrt
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.