FormalConjectures.ErdosProblems.«64»

/-
Copyright 2025 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

Erdős Problem 64

Reference: erdosproblems.com/64


namespace Erdos64

Does every finite graph with minimum degree at least $3$ contain a cycle of length $2^k$ for some $k \geq 2$?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_64 :
    answer(sorry) ↔ ∀ (V : Type*) (G : SimpleGraph V) [Fintype V] [DecidableRel G.Adj],
        G.minDegree ≥ 3 → ∃ (k : ℕ) (v : V) (c : G.Walk v v),
            k ≥ 2 ∧ c.IsCycle ∧ c.length = 2^k := by⊢ True ↔
  ∀ (V : Type u_1) (G : SimpleGraph V) [inst : Fintype V] [inst_1 : DecidableRel G.Adj],
    G.minDegree ≥ 3 → ∃ k v c, k ≥ 2 ∧ c.IsCycle ∧ c.length = 2 ^ k
  sorryAll goals completed! 🐙


-- TODO(firsching): add more context

end Erdos64