FormalConjectures.ErdosProblems.«609»

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module

public import FormalConjecturesUtil

Erdős Problem 609

References:

erdosproblems.com/609

[ErGr75] Erdős, P. and Graham, R. L., On partition theorems for finite graphs. Colloq. Math. Soc. János Bolyai (1975).

[Ch97] Chung, F., Open problems of Paul Erdős in graph theory. J. Graph Theory (1997), 3-36.

[DaJo17] Day, A. N. and Johnson, J. R., Multicolour Ramsey numbers of odd cycles. J. Combin. Theory Ser. B (2017), 56-63.

[GiHu24] Girão, A. and Hunter, Z., Monochromatic odd cycles in edge-coloured complete graphs. arXiv:2412.07708 (2024).

[JaYi25] Janzer, O. and Yip, F., Short monochromatic odd cycles. arXiv:2506.14910 (2025).

@[expose] public sectionopen Filternamespace Erdos609

The monochromatic subgraph of color i under edge coloring c : Sym2 V → Fin n.

def monochromaticGraph {V : Type*} {n : ℕ} (c : Sym2 V → Fin n) (i : Fin n) : SimpleGraph V :=
  SimpleGraph.fromRel (fun u v ↦ c s(u, v) = i)

A coloring c has a monochromatic odd cycle of length at most m.

def MonochromaticHasOddCycleLe (n : ℕ) (m : ℕ) (c : Sym2 (Fin (2 ^ n + 1)) → Fin n) : Prop :=
  ∃ (i : Fin n) (l : ℕ), l ∈ (monochromaticGraph c i).oddCycleLengths ∧ l ≤ m

$f(n)$ is the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$.

noncomputable def f (n : ℕ) : ℕ :=
  sInf {m : ℕ | ∀ (c : Sym2 (Fin (2 ^ n + 1)) → Fin n), MonochromaticHasOddCycleLe n m c}

Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$. Estimate $f(n)$.

@[category research open, AMS 5]
theorem erdos_609 :
    (fun n ↦ (f n : ℝ)) =Θ[atTop] (answer(sorry) : ℕ → ℝ) := by⊢ (fun n ↦ ↑(f n)) =Θ[atTop] sorry
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos609