FormalConjectures.ErdosProblems.«1068»

/-
Copyright 2026 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 1068

Reference: erdosproblems.com/1068


open Cardinal SimpleGraph


namespace Erdos1068

Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_1068 : answer(sorry) ↔
    ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = ℵ_  1 →
      ∃ s : Set V, s.Countable ∧ InfinitelyConnected (G.induce s) := by⊢ True ↔
  ∀ (V : Type) (G : SimpleGraph V), G.chromaticCardinal = ℵ_ 1 → ∃ s, s.Countable ∧ (induce s G).InfinitelyConnected
  sorryAll goals completed! 🐙


end Erdos1068