FormalConjectures.ErdosProblems.«923»

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import FormalConjecturesUtil

Erdős Problem 923

References:

erdosproblems.com/923

[Er69b] Erdős, P., Problems and results in chromatic graph theory. Proof Techniques in Graph Theory (1969), 27-35.

[Ro77] Rödl, V., On the chromatic number of subgraphs of a given graph. Proc. Amer. Math. Soc. (1977), 370-371.


namespace Erdos923


open SimpleGraph

Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number $\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\geq k$?

This is true, as shown by Rödl [Ro77].

@[category research solved, AMS 5, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos923.lean"]
theorem declaration uses 'sorry'erdos_923 : answer(True) ↔ ∀ (V : Type*) (n : ℕ),
    ∃ k : ℕ, ∀ G : SimpleGraph V, k ≤ G.chromaticNumber →
    ∃ H ≤ G, n ≤ H.chromaticNumber ∧ H.CliqueFree 3 := by⊢ True ↔ ∀ (V : Type u_1) (n : ℕ), ∃ k, ∀ (G : SimpleGraph V), ↑k ≤ χ(G) → ∃ H ≤ G, ↑n ≤ χ(H) ∧ H.CliqueFree 3
  sorryAll goals completed! 🐙


end Erdos923