FormalConjectures.ErdosProblems.«551»

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module

public import FormalConjecturesUtil

Erdős Problem 551

References:

erdosproblems.com/551

[BoEr73] Bondy, J. A. and Erdős, P., Ramsey numbers for cycles in graphs. J. Combin. Theory Ser. B (1973), 46-54.

[Ni05] Nikiforov, V., The cycle-complete graph Ramsey numbers. Combin. Probab. Comput. (2005), 349-370.

[KLS21] Keevash, P., Long, E. and Skokan, J., Cycle-complete Ramsey numbers. Int. Math. Res. Not. IMRN (2021), 277-302.

@[expose] public sectionopen Filternamespace Erdos551

Prove that $$R(C_k,K_n)=(k-1)(n-1)+1$$ for $k\geq n\geq 3$ (except when $n=k=3$).

Asked by Erdős, Faudree, Rousseau, and Schelp. This problem is #18 in Ramsey Theory in the graphs problem collection.

@[category research open, AMS 5]
theorem erdos_551 :
    ∀ (k n : ℕ), 3 ≤ n → n ≤ k → ¬(n = 3 ∧ k = 3) →
      SimpleGraph.graphRamsey (SimpleGraph.cycleGraph k)
        (SimpleGraph.completeGraph (Fin n)) = (k - 1) * (n - 1) + 1 := by⊢ ∀ (k n : ℕ),
  3 ≤ n →
    n ≤ k →
      ¬(n = 3 ∧ k = 3) →
        (SimpleGraph.cycleGraph k).graphRamsey (SimpleGraph.completeGraph (Fin n)) = (k - 1) * (n - 1) + 1
  sorryAll goals completed! 🐙

For sufficiently large $n$ and every $k\geq n$, $R(C_k,K_n)=(k-1)(n-1)+1$.

Keevash, Long, and Skokan [KLS21] have proved this identity when $k\geq C\frac{\log n}{\log\log n}$ for some constant $C$, thus establishing the conjecture for sufficiently large $n$.

@[category research solved, AMS 5]
theorem erdos_551.variants.sufficiently_large :
    ∀ᶠ n : ℕ in atTop, ∀ k : ℕ, n ≤ k →
      SimpleGraph.graphRamsey (SimpleGraph.cycleGraph k)
        (SimpleGraph.completeGraph (Fin n)) = (k - 1) * (n - 1) + 1 := by⊢ ∀ᶠ (n : ℕ) in atTop,
  ∀ (k : ℕ), n ≤ k → (SimpleGraph.cycleGraph k).graphRamsey (SimpleGraph.completeGraph (Fin n)) = (k - 1) * (n - 1) + 1
  sorryAll goals completed! 🐙

end Erdos551