742.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, 4 contain sorry, 0 declared as axioms

Erdős Problem 742

References:

erdosproblems.com/742
[Pl75] Plesník, Ján, Critical graphs of given diameter. Acta Fac. Rerum Natur. Univ. Comenian. Math. 30 (1975), 71-93.
[CaHa79] Caccetta, L. and Häggkvist, R., On diameter critical graphs. Discrete Math. 28 (1979), 223-229.
[Fa87] Fan, Genghua, On diameter 2-critical graphs. Discrete Math. 67 (1987), 235-240.
[Fü92] Füredi, Zoltán, The maximum number of edges in a minimal graph of diameter 2. J. Graph Theory 16 (1992), 81-98.

Imports (1)

FormalConjecturesUtil

def Erdos742.IsDiameter2Critical A graph is diameter-2-critical if it has diameter $2$ and removing any edge increases the diameter beyond $2$ line 44 source
Source documentation
A graph is diameter-2-critical if it has diameter $2$ and removing any edge increases the diameter beyond $2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph
Proof uses: And, Eq, Nat, SimpleGraph.diam, OfNat.ofNat, Sym2, Membership.mem, Set, SimpleGraph.edgeSet, Ne, SimpleGraph.deleteEdges, Singleton.singleton
3 instances, coercions and auxiliary constants left out.
```
def IsDiameter2Critical (G : SimpleGraph V) : Prop :=
…
```
theorem Erdos742.erdos_742 Murty-Simon Conjecture Let $G$ be a graph on $n$ vertices with diameter $2$ such that deleting any edge increases the diameter sorry line 58 source
Source documentation
Murty-Simon Conjecture
Let $G$ be a graph on $n$ vertices with diameter $2$ such that deleting any edge increases the diameter. Is it true that $G$ has at most $\lfloor n^2 / 4 \rfloor$ edges? Equality is conjectured to hold for the complete balanced bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor}$.
The conjecture is resolved up to a finite check: Fan [Fa87] verified it for $n \leq 24$ and $n = 26$, and Füredi [Fü92] proved it for all sufficiently large $n$.

depends on sorry what this means
Statement uses: Iff, True, Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Erdos742.IsDiameter2Critical, LE.le, Nat, Finset.card, Sym2, SimpleGraph.edgeFinset, HDiv.hDiv, HPow.hPow, Fintype.card, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_742 :
    answer(sorry) ↔ ∀ (V : Type*) [Fintype V] [DecidableEq V]
    (G : SimpleGraph V) [DecidableRel G.Adj], IsDiameter2Critical G →
    G.edgeFinset.card ≤ (Fintype.card V) ^ 2 / 4 := by
…
```
Mentions: Erdos742.IsDiameter2Critical
theorem Erdos742.complete_bipartite_edge_count The complete bipartite graph $K_{a, b}$ has exactly $a \cdot b$ edges line 70 source
Source documentation
The complete bipartite graph $K_{a, b}$ has exactly $a \cdot b$ edges. The bound $\lfloor n^2 / 4 \rfloor$ in the Murty-Simon conjecture is attained by the balanced case $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Eq, Set.ncard, Sym2, Sum, Fin, SimpleGraph.edgeSet, completeBipartiteGraph, HMul.hMul
Proof uses: Prod, Sym2.mk, Sum.inl, Prod.fst, Sum.inr, Prod.snd, Set, Set.image, Set.univ, Set.ext, Eq.mpr, Iff, Membership.mem, Exists, id, congrArg, Eq.trans, And, congrFun', Set.image_congr and 49 more
23 instances, coercions and auxiliary constants left out.

Tactic trace: 26 steps
Open to load the checked before-and-after goal states.
```
theorem complete_bipartite_edge_count (a b : ℕ) :
    (completeBipartiteGraph (Fin a) (Fin b)).edgeSet.ncard = a * b := by
…
```
theorem Erdos742.variants.plesnik_bound Plesník [Pl75] proved the bound $|E(G)| < 3n(n-1)/8$ for any diameter-$2$-critical graph on $n$ vertices sorry line 103 source
Source documentation
Plesník [Pl75] proved the bound $|E(G)| < 3n(n-1)/8$ for any diameter-$2$-critical graph on $n$ vertices.

depends on sorry what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Erdos742.IsDiameter2Critical, LT.lt, Real, Finset.card, Sym2, SimpleGraph.edgeFinset, HDiv.hDiv, HMul.hMul, OfNat.ofNat, Nat, Fintype.card, HSub.hSub
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
36 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem plesnik_bound (G : SimpleGraph V) [DecidableRel G.Adj] (hG : IsDiameter2Critical G) :
    (G.edgeFinset.card : ℝ) < 3 * (Fintype.card V : ℝ) * ((Fintype.card V : ℝ) - 1) / 8 := by
…
```
Mentions: Erdos742.IsDiameter2Critical
theorem Erdos742.variants.fan_bound Fan [Fa87] verified the Murty-Simon conjecture for all $n \leq 24$ and for $n = 26$ sorry line 111 source
Source documentation
Fan [Fa87] verified the Murty-Simon conjecture for all $n \leq 24$ and for $n = 26$.

depends on sorry what this means
Statement uses: Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, Or, LE.le, Nat, Fintype.card, OfNat.ofNat, Eq, Erdos742.IsDiameter2Critical, Finset.card, Sym2, SimpleGraph.edgeFinset, HDiv.hDiv, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem fan_bound (G : SimpleGraph V) [DecidableRel G.Adj]
    (hn : Fintype.card V ≤ 24 ∨ Fintype.card V = 26) (hG : IsDiameter2Critical G) :
    G.edgeFinset.card ≤ (Fintype.card V) ^ 2 / 4 := by
…
```
Mentions: Erdos742.IsDiameter2Critical
theorem Erdos742.variants.furedi_bound sorry line 123 source
depends on sorry what this means
Statement uses: Exists, Nat, Fintype, DecidableEq, SimpleGraph, DecidableRel, SimpleGraph.Adj, LE.le, Fintype.card, Erdos742.IsDiameter2Critical, Finset.card, Sym2, SimpleGraph.edgeFinset, HDiv.hDiv, HPow.hPow, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

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