FormalConjectures.ErdosProblems.«63»

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module

public import FormalConjecturesUtil

Erdős Problem 63

References:

erdosproblems.com/63

[dBEr51] de Bruijn, N. G. and Erdős, P., A colour problem for infinite graphs and a problem in the theory of relations. Indag. Math. (1951), 369--373.

[ErHa66] Erdős, P. and Hajnal, A., On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. (1966), 61-99.

[LiMo20] Liu, Hong and Montgomery, Richard, A solution to Erdős and Hajnal's odd cycle problem. arXiv:2010.15802 (2020).

[Re24] Reiher, C., Graphs of large girth. arXiv:2403.13571 (2024).

@[expose] public sectionnamespace Erdos63

Does every graph with infinite chromatic number contain a cycle of length $2^n$ for infinitely many $n$?

Conjectured by Mihók and Erdős. Solved affirmatively following the work of Liu and Montgomery [LiMo20].

The linked formal proof (Codex) states the conclusion as {n | HasCycleLength G (2 ^ n)}.Infinite, with HasCycleLength G m unfolding to m ∈ G.cycleLengths.

@[category research solved, AMS 5, formal_proof using lean4 at
  "https://github.com/plby/lean-proofs/blob/8822f7ddef30fadbd92e1c6ab4ed897af356af5e/src/latest/ErdosProblems/Erdos63.lean#L46"]
theorem erdos_63 :
    answer(True) ↔
      ∀ {V : Type*} (G : SimpleGraph V), G.chromaticNumber = ⊤ →
        ∀ N : ℕ, ∃ n ≥ N, 2 ^ n ∈ G.cycleLengths := by⊢ True ↔ ∀ {V : Type u_1} (G : SimpleGraph V), G.chromaticNumber = ⊤ → ∀ (N : ℕ), ∃ n ≥ N, 2 ^ n ∈ G.cycleLengths
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos63