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import FormalConjecturesUtilErdős Problem 613
namespace Erdos613
Erdős Problem 613: Let $n \geq 3$ and $G$ be a graph with $\binom{2n+1}{2} - \binom{n}{2} - 1$ edges. Must $G$ be the union of a bipartite graph and a graph with maximum degree less than $n$?
@[category research solved, AMS 5]
theorem erdos_613 :
answer(False) ↔
∀ n ≥ 3, ∀ (V : Type*) [Fintype V] (G : SimpleGraph V), [DecidableRel G.Adj] →
G.edgeFinset.card = Nat.choose (2 * n + 1) 2 - Nat.choose n 2 - 1 →
∃ (B D : SimpleGraph V), [DecidableRel B.Adj] → [DecidableRel D.Adj] →
G = B ⊔ D ∧ B.IsBipartite ∧ ∀ v, D.degree v < n := ⊢ False ↔
∀ n ≥ 3,
∀ (V : Type u_1) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj],
G.edgeFinset.card = (2 * n + 1).choose 2 - n.choose 2 - 1 →
∃ B D,
∀ [DecidableRel B.Adj] [inst_3 : DecidableRel D.Adj], G = B ⊔ D ∧ B.IsBipartite ∧ ∀ (v : V), D.degree v < n
All goals completed! 🐙
end Erdos613