609.lean`

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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

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 609

References:

erdosproblems.com/609
[ErGr75] Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Colloq. Math. Soc. János Bolyai (1975).
[Ch97] Chung, F., Open problems of Paul Erdős in graph theory. J. Graph Theory (1997), 3-36.
[DaJo17] Day, A. N. and Johnson, J. R., Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B (2017), 56-63.
[GiHu24] Girão, A. and Hunter, Z., Monochromatic odd cycles in edge-coloured complete graphs. arXiv:2412.07708 (2024).
[JaYi25] Janzer, O. and Yip, F., Short monochromatic odd cycles. arXiv:2506.14910 (2025).

Imports (1)

FormalConjecturesUtil

def Erdos609.monochromaticGraph The monochromatic subgraph of color i under edge coloring c : Sym2 V → Fin n line 45 source
Source documentation
The monochromatic subgraph of color i under edge coloring c : Sym2 V → Fin n.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Sym2, Fin, SimpleGraph
Proof uses: SimpleGraph.fromRel, Eq, Sym2.mk
```
def monochromaticGraph {V : Type*} {n : ℕ} (c : Sym2 V → Fin n) (i : Fin n) : SimpleGraph V :=
…
```
Mentions: c
def Erdos609.MonochromaticHasOddCycleLe A coloring c has a monochromatic odd cycle of length at most m line 51 source
Source documentation
A coloring c has a monochromatic odd cycle of length at most m.

standard axioms: Quot.sound, propext what this means
Statement uses: Nat, Sym2, Fin, HAdd.hAdd, HPow.hPow, OfNat.ofNat
Proof uses: Exists, And, Membership.mem, Set, SimpleGraph.oddCycleLengths, Erdos609.monochromaticGraph, LE.le
16 instances, coercions and auxiliary constants left out.
```
def MonochromaticHasOddCycleLe (n : ℕ) (m : ℕ) (c : Sym2 (Fin (2 ^ n + 1)) → Fin n) : Prop :=
…
```
Mentions: c
def Erdos609.f $f(n)$ is the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$ line 58 source
Source documentation
$f(n)$ is the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Sym2, Fin, HAdd.hAdd, HPow.hPow, OfNat.ofNat, Erdos609.MonochromaticHasOddCycleLe
8 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos609.erdos_609 Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$ sorry line 66 source
Source documentation
Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Erdos609.f, sorryAx, Bool.false
Proof uses: Lean.Name, 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.
```
theorem erdos_609 :
    (fun n ↦ (f n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by
…
```
Mentions: Erdos609.f