566.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 566

References:

erdosproblems.com/566
[EFRS93] Erdős, Faudree, Rousseau and Schelp, Ramsey size linear graphs. Combin. Probab. Comput. (1993), 389-399.

Imports (1)

FormalConjecturesUtil

theorem Erdos566.erdos_566 Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges sorry line 43 source
Source documentation
Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $R(G,H) \ll m$?
In other words: if $G$ is sparse (every induced subgraph on $k$ vertices has $≤ 2k-3$ edges), is $G$ Ramsey size linear?

depends on sorry what this means
Statement uses: Iff, True, Nat, SimpleGraph, Fin, Finset, LE.le, OfNat.ofNat, Finset.card, Set.ncard, Sym2, SimpleGraph.edgeSet, SimpleGraph.induce, HSub.hSub, HMul.hMul, SimpleGraph.IsRamseySizeLinear
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_566 : answer(sorry) ↔
    ∀ (p : ℕ) (G : SimpleGraph (Fin p)),
      (∀ S : Finset (Fin p), 2 ≤ S.card → (G.induce S).edgeSet.ncard ≤ 2 * S.card - 3) →
      G.IsRamseySizeLinear := by
…
```