136.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

4 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 136

References:

erdosproblems.com/136
[Er97b] Erdős, Paul, Some old and new problems in various branches of combinatorics. Discrete Math. (1997), 227-231.
[BCDP22] Bennett, P. and Cushman, R. and Dudek, A. and Pralat, P., The Erdős-Gyárfás function $f(n,4,5)=\frac{5}{6}n+o(n)$ - so Gyárfás was right. arXiv:2207.02920 (2022).
[JoMu22] Joos, F. and Mubayi, D., Ramsey theory constructions from hypergraph matchings. arXiv:2208.12563 (2022).

Imports (1)

FormalConjecturesUtil

def Erdos136.Is45Coloring An edge colouring of K_n with k colours is a (4, 5)-colouring if the six edges of every copy of K_4 receive at least five distinct colours line 43 source
Source documentation
An edge colouring of K_n with k colours is a (4, 5)-colouring if the six edges of every copy of K_4 receive at least five distinct colours.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, SimpleGraph.TopEdgeLabeling, Fin
Proof uses: Function.Embedding, OfNat.ofNat, LE.le, Finset.card, Finset.image, Sym2, SimpleGraph.edgeSet, Top.top, SimpleGraph, SimpleGraph.TopEdgeLabeling.pullback, Finset.univ
12 instances, coercions and auxiliary constants left out.
```
def Is45Coloring {n k : ℕ} (C : SimpleGraph.TopEdgeLabeling (Fin n) (Fin k)) : Prop :=
…
```
def Erdos136.Colorable K_n admits a (4, 5)-colouring with k colours line 48 source
Source documentation
K_n admits a (4, 5)-colouring with k colours.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, SimpleGraph.TopEdgeLabeling, Fin, Erdos136.Is45Coloring
```
def Colorable (n k : ℕ) : Prop :=
…
```
def Erdos136.f f n is the least number of colours in a (4, 5)-colouring of K_n line 52 source
Source documentation
f n is the least number of colours in a (4, 5)-colouring of K_n.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, Erdos136.Colorable
1 instances, coercions and auxiliary constants left out.
```
noncomputable def f (n : ℕ) : ℕ :=
…
```
theorem Erdos136.erdos_136 sorry line 65 source
depends on sorry what this means
Statement uses: Filter.Tendsto, Nat, Real, HDiv.hDiv, Erdos136.f, Filter.atTop, nhds, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
26 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.