FormalConjectures.ErdosProblems.«73»

/-
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.
-/
module

public import FormalConjecturesUtil

Erdős Problem 73

References:

erdosproblems.com/73

[Re99] Reed, B., Mangoes and Blueberries. Combinatorica (1999), 267-296.

@[expose] public sectionnamespace Erdos73

Let $k\ge 0$. Let $G$ be a graph such that every subgraph $H$ contains an independent set of size $\ge (n-k)/2$, where $n$ is the number of vertices of $H$. Must $G$ be the union of a bipartite graph and $O_k(1)$ many vertices?

Proved by Reed [Re99].

@[category research solved, AMS 5]
theorem erdos_73 : answer(True) ↔
    ∀ (k : ℕ), ∃ (C : ℕ),
      ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
        (∀ (S : Finset V), ∃ (I : Finset V), I ⊆ S ∧ (G.induce (I : Set V)).edgeSet = ∅ ∧
          (I.card : ℝ) ≥ (S.card - k : ℝ) / 2) →
        ∃ (D : Finset V), D.card ≤ C ∧
          (G.induce (D : Set V)ᶜ).Colorable 2 := by⊢ True ↔
  ∀ (k : ℕ),
    ∃ C,
      ∀ (V : Type) [Fintype V] (G : SimpleGraph V),
        (∀ (S : Finset V), ∃ I ⊆ S, (SimpleGraph.induce (↑I) G).edgeSet = ∅ ∧ ↑I.card ≥ (↑S.card - ↑k) / 2) →
          ∃ D, D.card ≤ C ∧ (SimpleGraph.induce (↑D)ᶜ G).Colorable 2
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos73