567.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

6 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 567

Let $G$ be either $Q_3$ or $K_{3,3}$ or $H_5$ (the last formed by adding two vertex-disjoint chords to $C_5$). Is it true that, if $H$ has $m$ edges and no isolated vertices, then $$ R(G,H) \ll m? $$

In other words, is $G$ Ramsey size linear? A special case of Problem 566.

Reference: erdosproblems.com/567

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

Imports (1)

FormalConjecturesUtil

abbrev Erdos567.Q3 $Q_3$ is the 3-dimensional hypercube graph (8 vertices, 12 edges) line 44 source
Source documentation
$Q_3$ is the 3-dimensional hypercube graph (8 vertices, 12 edges). Vertices are 3-bit vectors. Two vertices are adjacent iff they differ in exactly one bit.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Fin, OfNat.ofNat, Nat, Bool
Proof uses: SimpleGraph.hypercube
2 instances, coercions and auxiliary constants left out.
```
abbrev Q3 : SimpleGraph (Fin 3 → Bool) := hypercube 3
```
def Erdos567.K33 $K_{3,3}$ is the complete bipartite graph with partition sizes 3, 3 (6 vertices, 9 edges) line 47 source
Source documentation
$K_{3,3}$ is the complete bipartite graph with partition sizes 3, 3 (6 vertices, 9 edges).

standard axioms: Quot.sound, propext what this means
Statement uses: SimpleGraph, Sum, Fin, OfNat.ofNat, Nat
Proof uses: completeBipartiteGraph
2 instances, coercions and auxiliary constants left out.
```
def K33 : SimpleGraph (Fin 3 ⊕ Fin 3) := completeBipartiteGraph (Fin 3) (Fin 3)
```
def Erdos567.H5 $H_5$ is $C_5$ with two vertex-disjoint chords (5 vertices, 7 edges) line 51 source
Source documentation
$H_5$ is $C_5$ with two vertex-disjoint chords (5 vertices, 7 edges). Also known as $K_4^*$ (the graph obtained from $K_4$ by subdividing one edge).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Fin, OfNat.ofNat, Nat
Proof uses: Max.max, SimpleGraph.cycleGraph, SimpleGraph.edge
5 instances, coercions and auxiliary constants left out.
```
def H5 : SimpleGraph (Fin 5) :=
…
```
theorem Erdos567.erdos_567.parts.i Erdős Problem 567 (Q3) Is $Q_3$ (the 3-dimensional hypercube) Ramsey size linear? sorry line 60 source
Source documentation
Erdős Problem 567 (Q3)
Is $Q_3$ (the 3-dimensional hypercube) Ramsey size linear?

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph.IsRamseySizeLinear, Fin, OfNat.ofNat, Nat, Bool, Erdos567.Q3
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_567.parts.i : answer(sorry) ↔ IsRamseySizeLinear Q3 := by
…
```
Mentions: Erdos567.Q3
theorem Erdos567.erdos_567.parts.ii Erdős Problem 567 (K33) Is $K_{3,3}$ Ramsey size linear? sorry line 69 source
Source documentation
Erdős Problem 567 (K33)
Is $K_{3,3}$ Ramsey size linear?

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph.IsRamseySizeLinear, Sum, Fin, OfNat.ofNat, Nat, Erdos567.K33
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_567.parts.ii : answer(sorry) ↔ IsRamseySizeLinear K33 := by
…
```
Mentions: Erdos567.K33
theorem Erdos567.erdos_567.parts.iii Erdős Problem 567 (H5) Is $H_5$ ($C_5$ with two vertex-disjoint chords) Ramsey size linear? sorry line 78 source
Source documentation
Erdős Problem 567 (H5)
Is $H_5$ ($C_5$ with two vertex-disjoint chords) Ramsey size linear?

depends on sorry what this means
Statement uses: Iff, True, SimpleGraph.IsRamseySizeLinear, Fin, OfNat.ofNat, Nat, Erdos567.H5
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_567.parts.iii : answer(sorry) ↔ IsRamseySizeLinear H5 := by
…
```
Mentions: Erdos567.H5