60.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 60

References:

erdosproblems.com/60
[HeMaYa21] He, J. and Ma, J. and Yang, T., Some extremal results on 4-cycles. Journal of Combinatorial Theory B (2021).

Imports (1)

FormalConjecturesUtil

theorem Erdos60.erdos_60 Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$? sorry line 41 source
Source documentation
Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, LT.lt, SimpleGraph.extremalNumber, SimpleGraph.cycleGraph, Finset.card, Sym2, SimpleGraph.edgeFinset, LE.le, HMul.hMul, Real.sqrt and 8 more
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_60 :
    ∃ c : ℝ, c > 0 ∧
      ∀ᶠ n : ℕ in atTop,
        ∀ (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
          (extremalNumber n (cycleGraph 4) < G.edgeFinset.card) →
          (c * Real.sqrt (n : ℝ) ≤ ({ H' : G.Subgraph | Nonempty (H'.coe ≃g cycleGraph 4) }.ncard : ℝ)) := by
…
```
Mentions: c
theorem Erdos60.erdos_60.variants.he_ma_yang Documented declaration: he_ma_yang sorry line 58 source
Source documentation
He, Ma, and Yang [HeMaYa21] proved the conjecture when $n = q^2 + q + 1$ for some $q = 2^k$: they showed that for large even $q$ every graph on $q^2 + q + 1$ vertices with $\frac{1}{2}q(q+1)^2 + 1$ edges contains at least $q - 1$ copies of $C_4$, and $\mathrm{ex}(q^2 + q + 1; C_4) = \frac{1}{2}q(q+1)^2$ when $q = 2^k$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Nat, Nat.isPowerOfTwo, SimpleGraph, Fin, HAdd.hAdd, HPow.hPow, DecidableRel, SimpleGraph.Adj, LT.lt, SimpleGraph.extremalNumber, SimpleGraph.cycleGraph, Finset.card, Sym2, SimpleGraph.edgeFinset, LE.le and 9 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
48 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_60.variants.he_ma_yang :
    ∃ c : ℝ, c > 0 ∧
      ∀ (q : ℕ) (_hq : q.isPowerOfTwo),
        ∀ (G : SimpleGraph (Fin (q^2 + q + 1))) [DecidableRel G.Adj],
          (extremalNumber (q^2 + q + 1) (cycleGraph 4) < G.edgeFinset.card) →
          (c * Real.sqrt ((q^2 + q + 1) : ℝ) ≤ ({ H' : G.Subgraph | Nonempty (H'.coe ≃g cycleGraph 4) }.ncard : ℝ)) := by
…
```
Mentions: c
theorem Erdos60.erdos_60.variants.two_copies Erdős and Simonovits conjectured that at least 2 copies of $C_4$ are guaranteed sorry line 70 source
Source documentation
Erdős and Simonovits conjectured that at least 2 copies of $C_4$ are guaranteed.

depends on sorry what this means
Statement uses: Filter.Eventually, Nat, SimpleGraph, Fin, DecidableRel, SimpleGraph.Adj, LT.lt, SimpleGraph.extremalNumber, OfNat.ofNat, SimpleGraph.cycleGraph, Finset.card, Sym2, SimpleGraph.edgeFinset, LE.le, Set.ncard, SimpleGraph.Subgraph, Set.ofPred, Nonempty, SimpleGraph.Iso, SimpleGraph.Subgraph.verts and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_60.variants.two_copies :
    ∀ᶠ n : ℕ in atTop,
      ∀ (G : SimpleGraph (Fin n)) [DecidableRel G.Adj],
        (extremalNumber n (cycleGraph 4) < G.edgeFinset.card) →
        (2 ≤ { H' : G.Subgraph | Nonempty (H'.coe ≃g cycleGraph 4) }.ncard) := by
…
```