Erdős Problem 1080

Reference: erdosproblems.com/1080

def Erdos1080.IsBipartition {V : Type u_1} (G : SimpleGraph V) (X Y : Set V) : Prop

IsBipartition G X Y means that X and Y form a bipartition of the vertices of G.

Equations

• Erdos1080.IsBipartition G X Y = (Disjoint X Y ∧ X ∪ Y = Set.univ ∧ ∀ ⦃u v : V⦄, G.Adj u v → (u ∈ X ↔ v ∈ Y))

theorem Erdos1080.erdos_1080 : sorry ↔ ∃ c > 0, ∀ (V : Type) [inst : Fintype V] [Nonempty V] (G : SimpleGraph V) (X Y : Set V), IsBipartition G X Y → X.ncard = ⌊↑(Fintype.card V) ^ (2 / 3)⌋₊ → ↑G.edgeSet.ncard ≥ c * ↑(Fintype.card V) → ∃ (v : V) (walk : G.Walk v v), walk.IsCycle ∧ walk.length = 6

Let $G$ be a bipartite graph on $n$ vertices such that one part has $\lfloor n^{2/3}\rfloor$ vertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must contain a $C_6$?