FormalConjectures.ErdosProblems.«57»

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module

public import FormalConjecturesUtil

Erdős Problem 57

References:

erdosproblems.com/57

[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).

@[expose] public sectionnamespace Erdos57

If $G$ is a graph with infinite chromatic number and $a_1 < a_2 < \cdots$ are lengths of the odd cycles of $G$ then $\sum \frac{1}{a_i} = \infty$.

Conjectured by Erdős and Hajnal [ErHa66], and solved by Liu and Montgomery [LiMo20].

The linked formal proof (Codex and GPT-5.6 Sol) states the conclusion as ¬ Summable (oddCycleReciprocal G), where oddCycleReciprocal G n is n⁻¹ if n is an odd cycle length of G and 0 otherwise; this is the indicator form of the sum below.

@[category research solved, AMS 5, formal_proof using lean4 at
  "https://github.com/plby/lean-proofs/blob/8822f7ddef30fadbd92e1c6ab4ed897af356af5e/src/latest/ErdosProblems/Erdos57.lean#L11124"]
theorem erdos_57 :
    ∀ {V : Type*} (G : SimpleGraph V), G.chromaticNumber = ⊤ →
      ¬ Summable (fun (a : G.oddCycleLengths) ↦ 1 / (a : ℝ)) := by⊢ ∀ {V : Type u_1} (G : SimpleGraph V), G.chromaticNumber = ⊤ → ¬Summable fun a ↦ 1 / ↑↑a
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos57