180.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

7 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 180

References:

erdosproblems.com/180
[ErSi82] Erdős, P. and Simonovits, M., Compactness results in extremal graph theory. Combinatorica (1982), 275-288.
[OpenAI26] OpenAI, Ten advances in mathematics and theoretical computer science. (2026).

Imports (1)

FormalConjecturesUtil

structure Erdos180.FiniteGraph A finite graph, bundled with its vertex count, so that a family may mix orders line 37 source
Source documentation
A finite graph, bundled with its vertex count, so that a family may mix orders.

no axioms what this means
```
structure FiniteGraph where
…
```
def Erdos180.FamilyFree A host graph is family-free when it contains no member of family as a subgraph line 42 source
Source documentation
A host graph is family-free when it contains no member of family as a subgraph.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Erdos180.FiniteGraph, Nat, SimpleGraph, Fin
Proof uses: Membership.mem, SimpleGraph.Free, Erdos180.FiniteGraph.order, Erdos180.FiniteGraph.graph
2 instances, coercions and auxiliary constants left out.
```
def FamilyFree (family : Finset FiniteGraph) {n : ℕ} (host : SimpleGraph (Fin n)) : Prop :=
…
```
Mentions: Erdos180.FiniteGraph
def Erdos180.familyExtremal $\mathrm{ex}(n;\mathcal{F})$, the greatest number of edges of a graph on n vertices containing no member of family line 48 source
Source documentation
$\mathrm{ex}(n;\mathcal{F})$, the greatest number of edges of a graph on n vertices containing no member of family.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Erdos180.FiniteGraph, Nat
Proof uses: Finset.sup, SimpleGraph, Fin, Finset.filter, Erdos180.FamilyFree, Classical.propDecidable, Finset.univ, Finset.card, Sym2, SimpleGraph.edgeFinset, SimpleGraph.Adj
8 instances, coercions and auxiliary constants left out.
```
noncomputable def familyExtremal (family : Finset FiniteGraph) (n : ℕ) : ℕ :=
…
```
Mentions: Erdos180.FiniteGraph
def Erdos180.IsCyclicFamily No member of the family is acyclic line 53 source
Source documentation
No member of the family is acyclic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Erdos180.FiniteGraph
Proof uses: Membership.mem, Not, SimpleGraph.IsAcyclic, Fin, Erdos180.FiniteGraph.order, Erdos180.FiniteGraph.graph
2 instances, coercions and auxiliary constants left out.
```
def IsCyclicFamily (family : Finset FiniteGraph) : Prop :=
…
```
Mentions: Erdos180.FiniteGraph
def Erdos180.IsCompactFamily The family is compact: some single member already controls the family extremal number line 57 source
Source documentation
The family is compact: some single member already controls the family extremal number.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Finset, Erdos180.FiniteGraph
Proof uses: Exists, And, Membership.mem, Real, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, SimpleGraph.extremalNumber, Fin, Erdos180.FiniteGraph.order, Erdos180.FiniteGraph.graph, HMul.hMul, Erdos180.familyExtremal, Filter.atTop
11 instances, coercions and auxiliary constants left out.
```
def IsCompactFamily (family : Finset FiniteGraph) : Prop :=
…
```
Mentions: Erdos180.FiniteGraph
theorem Erdos180.erdos_180 Documented declaration: erdos_180 sorry line 75 source
Source documentation
If $\mathcal{F}$ is a finite set of finite graphs then $\mathrm{ex}(n;\mathcal{F})$ is the maximum number of edges a graph on $n$ vertices can have without containing any subgraphs from $\mathcal{F}$. Note that it is trivial that $\mathrm{ex}(n;\mathcal{F})\leq \mathrm{ex}(n;G)$ for every $G\in\mathcal{F}$. Is it true that, for every $\mathcal{F}$, there exists $G\in\mathcal{F}$ such that $$\mathrm{ex}(n;G)\ll_{\mathcal{F}}\mathrm{ex}(n;\mathcal{F})?$$
This is the Erdős–Simonovits compactness conjecture. The answer is no: OpenAI [OpenAI26] give a family of connected bipartite graphs, none of them acyclic, for which no single member controls the family extremal number. See erdos_180.variants.counterexample.

depends on sorry what this means
Statement uses: Iff, False, Finset, Erdos180.FiniteGraph, Finset.Nonempty, Erdos180.IsCyclicFamily, Erdos180.IsCompactFamily
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_180 : answer(False) ↔
    ∀ family : Finset FiniteGraph,
      family.Nonempty → IsCyclicFamily family → IsCompactFamily family := by
…
```
Mentions: Erdos180.FiniteGraph, Erdos180.IsCyclicFamily, Erdos180.IsCompactFamily
theorem Erdos180.erdos_180.variants.counterexample sorry line 86 source
depends on sorry what this means
Statement uses: Exists, Finset, Erdos180.FiniteGraph, And, Finset.Nonempty, Membership.mem, SimpleGraph.Connected, Fin, Erdos180.FiniteGraph.order, Erdos180.FiniteGraph.graph, SimpleGraph.IsBipartite, Not, SimpleGraph.IsAcyclic, Erdos180.IsCompactFamily
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
5 instances, coercions and auxiliary constants left out.

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