24.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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 24

References:

erdosproblems.com/24
[Er90] Erdős, Paul, Some of my favourite unsolved problems. A tribute to Paul Erdős (1990), 467-478.
[Er97b] Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.
[Er92b] Erdős, Paul, Some of my favourite problems in various branches of combinatorics. Matematiche (Catania) (1992), 231-240.
[Er97f] Erdős, Paul, Some unsolved problems. Combinatorics, geometry and probability (Cambridge, 1993) (1997), 1-10.
[Gr12] Grzesik, Andrzej, On the maximum number of five-cycles in a triangle-free graph. J. Combin. Theory Ser. B (2012), 1061-1066.
[HHKNR13] Hatami, Hamed and Hladký, Jan and Kráľ, Daniel and Norine, Serguei and Razborov, Alexander, On the number of pentagons in triangle-free graphs. J. Combin. Theory Ser. A (2013), 722-732.

Imports (1)

FormalConjecturesUtil

theorem Erdos24.erdos_24 Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$? sorry line 53 source
Source documentation
Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?
Győri proved this with $1.03n^5$, which has been improved by Füredi. The answer is yes, as proved independently by Grzesik [Gr12] and Hatami, Hladky, Král, Norine, and Razborov [HHKNR13].

depends on sorry what this means
Statement uses: Iff, True, Nat, SimpleGraph, Fin, HMul.hMul, OfNat.ofNat, SimpleGraph.CliqueFree, LE.le, SimpleGraph.copyCount, SimpleGraph.cycleGraph, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
18 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_24 : answer(True) ↔
    ∀ (n : ℕ) (G : SimpleGraph (Fin (5 * n))), G.CliqueFree 3 →
      G.copyCount (cycleGraph 5) ≤ n ^ 5 := by
…
```