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import FormalConjecturesUtilErdős Problem 146
References:
[ErSi84] Erdős, P. and Simonovits, M., Cube-supersaturated graphs and related problems. Progress in graph theory (1984), 203-218.
[OpenAI26] OpenAI, Ten advances in mathematics and theoretical computer science. (2026).
open Filter SimpleGraphnamespace Erdos146open scoped Classical in
The neighbours of v lying inside s.
noncomputable def neighborsWithin {V : Type*} (H : SimpleGraph V) (s : Finset V) (v : V) :
Finset V := s.filter (H.Adj v)
H is r-degenerate when every induced subgraph has a vertex of degree at most r, that is,
every nonempty vertex set contains a vertex with at most r neighbours inside it.
def IsDegenerate {V : Type*} (r : ℕ) (H : SimpleGraph V) : Prop :=
∀ s : Finset V, s.Nonempty → ∃ v ∈ s, (neighborsWithin H s v).card ≤ rIf $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $$\mathrm{ex}(n;H) \ll n^{2-1/r}.$$
The answer is no. OpenAI [OpenAI26] give a connected bipartite 2-degenerate H and constants
c, ε > 0 with $\mathrm{ex}(n;H)\geq cn^{3/2+\epsilon}$ for all large n, which exceeds the
conjectured $n^{2-1/2}=n^{3/2}$. See erdos_146.variants.two_degenerate_counterexample.
@[category research solved, AMS 5]
theorem erdos_146 : answer(False) ↔
∀ (r q : ℕ) (H : SimpleGraph (Fin q)),
0 < r → H.IsBipartite → IsDegenerate r H →
Asymptotics.IsBigO atTop
(fun n : ℕ => (extremalNumber n H : ℝ))
(fun n : ℕ => (n : ℝ) ^ ((2 : ℝ) - 1 / (r : ℝ))) := ⊢ False ↔
∀ (r q : ℕ) (H : SimpleGraph (Fin q)),
0 < r → H.IsBipartite → IsDegenerate r H → (fun n ↦ ↑(extremalNumber n H)) =O[atTop] fun n ↦ ↑n ^ (2 - 1 / ↑r)
All goals completed! 🐙
The counterexample: a connected bipartite 2-degenerate H whose extremal number exceeds
$n^{3/2+\epsilon}$ infinitely often, so the r = 2 case of erdos_146 fails.
@[category research solved, AMS 5, formal_proof using lean4 at
"https://github.com/openai/ten-proofs/blob/94bc0feb6a9ff12c7d31d6de640a725c9d43d2b6/CompactnessAndDegeneracy.lean"]
theorem erdos_146.variants.two_degenerate_counterexample :
∃ (q : ℕ) (H : SimpleGraph (Fin q)),
H.Connected ∧ H.IsBipartite ∧ IsDegenerate 2 H ∧
∃ c ε : ℝ, 0 < c ∧ 0 < ε ∧
∀ᶠ n : ℕ in atTop,
c * (n : ℝ) ^ ((3 : ℝ) / 2 + ε) ≤ (extremalNumber n H : ℝ) := ⊢ ∃ q H,
H.Connected ∧
H.IsBipartite ∧
IsDegenerate 2 H ∧ ∃ c ε, 0 < c ∧ 0 < ε ∧ ∀ᶠ (n : ℕ) in atTop, c * ↑n ^ (3 / 2 + ε) ≤ ↑(extremalNumber n H)
All goals completed! 🐙end Erdos146