1036.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

5 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 1036

References:

erdosproblems.com/1036
[AlHa91] Alon, N. and Hajnal, A., Ramsey graphs contain many distinct induced subgraphs. Graphs Combin. (1991), 1-6.
[ErHa89b] Erdős, P. and Hajnal, A., On the number of distinct induced subgraphs of a graph. Discrete Math. (1989), 145-154.
[Sh98] Shelah, Saharon, Erdős and Rényi conjecture. J. Combin. Theory Ser. A (1998), 179-185.

Imports (1)

FormalConjecturesUtil

def Erdos1036.HasManyNonIsomorphicInducedSubgraphs Documented declaration: HasManyNonIsomorphicInducedSubgraphs line 43 source
Source documentation
G contains at least k many induced subgraphs which are not pairwise isomorphic, that is, there is a family of at least k many sets of vertices whose induced subgraphs are pairwise non-isomorphic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Real
Proof uses: Exists, Set, And, Set.Pairwise, IsEmpty, SimpleGraph.Iso, SimpleGraph.induce, LE.le, Set.ncard
4 instances, coercions and auxiliary constants left out.
```
def HasManyNonIsomorphicInducedSubgraphs {V : Type*} (G : SimpleGraph V) (k : ℝ) : Prop :=
…
```
def Erdos1036.NoLargeBiclique Neither G nor its complement contains a balanced complete bipartite graph $K_{k,k}$ on more than m vertices as a (not necessarily induced) subgraph line 53 source
Source documentation
Neither G nor its complement contains a balanced complete bipartite graph $K_{k,k}$ on more than m vertices as a (not necessarily induced) subgraph. That is, there are no two disjoint sets of k vertices with $2k > m$ such that every pair of vertices from different sets is adjacent in G, or every such pair is non-adjacent in G.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: SimpleGraph, Real
Proof uses: Nat, Or, SimpleGraph.IsContained, Sum, Fin, completeBipartiteGraph, Compl.compl, LE.le, HMul.hMul, OfNat.ofNat
8 instances, coercions and auxiliary constants left out.
```
def NoLargeBiclique {V : Type*} (G : SimpleGraph V) (m : ℝ) : Prop :=
…
```
theorem Erdos1036.erdos_1036 Let $G$ be a graph on $n$ vertices which does not contain a trivial (empty or complete) graph on more than $c\log n$ vertices sorry line 65 source
Source documentation
Let $G$ be a graph on $n$ vertices which does not contain a trivial (empty or complete) graph on more than $c\log n$ vertices. Must $G$ contain at least $2^{\Omega_c(n)}$ many induced subgraphs which are not pairwise isomorphic?
This is true, and was proved by Shelah [Sh98].

depends on sorry what this means
Statement uses: Iff, True, Real, LT.lt, OfNat.ofNat, Exists, And, Filter.Eventually, Nat, SimpleGraph, Fin, LE.le, SimpleGraph.cliqueNum, HMul.hMul, Real.log, SimpleGraph.indepNum, Erdos1036.HasManyNonIsomorphicInducedSubgraphs, HPow.hPow, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1036 : answer(True) ↔
    ∀ c : ℝ, 0 < c → ∃ δ : ℝ, 0 < δ ∧ ∀ᶠ n : ℕ in atTop, ∀ G : SimpleGraph (Fin n),
      (G.cliqueNum : ℝ) ≤ c * Real.log (n : ℝ) → (G.indepNum : ℝ) ≤ c * Real.log (n : ℝ) →
        HasManyNonIsomorphicInducedSubgraphs G ((2 : ℝ) ^ (δ * (n : ℝ))) := by
…
```
Mentions: c, Erdos1036.HasManyNonIsomorphicInducedSubgraphs
theorem Erdos1036.erdos_1036.variants.alon_hajnal Alon and Hajnal [AlHa91] proved that $G$ must contain at least $$\exp\left(n(\log n)^{-O(\log\log n)}\right)$$ many non-isomorphic induced subgraphs sorry line 77 source
Source documentation
Alon and Hajnal [AlHa91] proved that $G$ must contain at least $$\exp\left(n(\log n)^{-O(\log\log n)}\right)$$ many non-isomorphic induced subgraphs.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Exists, And, Filter.Eventually, Nat, SimpleGraph, Fin, LE.le, SimpleGraph.cliqueNum, HMul.hMul, Real.log, SimpleGraph.indepNum, Erdos1036.HasManyNonIsomorphicInducedSubgraphs, Real.exp, HPow.hPow, Neg.neg, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1036.variants.alon_hajnal :
    ∀ c : ℝ, 0 < c → ∃ C : ℝ, 0 < C ∧ ∀ᶠ n : ℕ in atTop, ∀ G : SimpleGraph (Fin n),
      (G.cliqueNum : ℝ) ≤ c * Real.log (n : ℝ) → (G.indepNum : ℝ) ≤ c * Real.log (n : ℝ) →
        HasManyNonIsomorphicInducedSubgraphs G
          (Real.exp ((n : ℝ) * Real.log (n : ℝ) ^ (-(C * Real.log (Real.log (n : ℝ)))))) := by
…
```
Mentions: c, Erdos1036.HasManyNonIsomorphicInducedSubgraphs
theorem Erdos1036.erdos_1036.variants.erdos_hajnal Erdős and Hajnal [ErHa89b] proved that if neither sorry line 90 source
Source documentation
Erdős and Hajnal [ErHa89b] proved that if neither $G$ nor its complement contains a balanced complete bipartite graph $K_{k,k}$ on more than $c\log n$ vertices (as a not necessarily induced subgraph) then $G$ contains at least $2^{\Omega_c(n)}$ many non-isomorphic induced subgraphs.

depends on sorry what this means
Statement uses: Real, LT.lt, OfNat.ofNat, Exists, And, Filter.Eventually, Nat, SimpleGraph, Fin, Erdos1036.NoLargeBiclique, HMul.hMul, Real.log, Erdos1036.HasManyNonIsomorphicInducedSubgraphs, HPow.hPow, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1036.variants.erdos_hajnal :
    ∀ c : ℝ, 0 < c → ∃ δ : ℝ, 0 < δ ∧ ∀ᶠ n : ℕ in atTop, ∀ G : SimpleGraph (Fin n),
      NoLargeBiclique G (c * Real.log (n : ℝ)) →
        HasManyNonIsomorphicInducedSubgraphs G ((2 : ℝ) ^ (δ * (n : ℝ))) := by
…
```
Mentions: c, Erdos1036.NoLargeBiclique, Erdos1036.HasManyNonIsomorphicInducedSubgraphs