71.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 71

References:

erdosproblems.com/71
[Bo77] Bollobás, Béla, Cycles modulo $k$. Bull. London Math. Soc. (1977), 97-98.
[Er82e] Erdős, Paul, Some of my favourite problems which recently have been solved. (1982), 59--79.
[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
[Er97b] Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.

Imports (1)

FormalConjecturesUtil

theorem Erdos71.erdos_71 Is it true that for every infinite arithmetic progression sorry line 53 source
Source documentation
Is it true that for every infinite arithmetic progression $P$ which contains even numbers there is some constant $c=c(P)$ such that every graph with average degree at least $c$ contains a cycle whose length is in $P$?
In [Er82e] Erdős credits this conjecture to himself and Burr. This has been proved by Bollobás [Bo77]. The best dependence of the constant $c(P)$ is unknown.
The infinite arithmetic progression is encoded as a set $P \subseteq \mathbb{N}$ satisfying P.IsAPOfLength ⊤ (which forces a positive common difference), and "contains even numbers" as the existence of an even element. The average degree of a finite simple graph is SimpleGraph.averageDegree, i.e. $(\sum_v \deg v)/|V| \in \mathbb{Q}$, and a cycle whose length is in $P$ is a cycle walk w with w.length ∈ P.

depends on sorry what this means
Statement uses: Iff, True, Set, Nat, Set.IsAPOfLength, Top.top, ENat, Exists, And, Membership.mem, Even, Rat, Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, LE.le, SimpleGraph.averageDegree, SimpleGraph.Walk and 2 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
11 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_71 : answer(True) ↔
    ∀ P : Set ℕ, P.IsAPOfLength ⊤ → (∃ n ∈ P, Even n) →
      ∃ c : ℚ, ∀ (V : Type) [Fintype V] [DecidableEq V] (G : SimpleGraph V)
        [DecidableRel G.Adj], c ≤ G.averageDegree →
          ∃ (v : V) (w : G.Walk v v), w.IsCycle ∧ w.length ∈ P := by
…
```
Mentions: c