FormalConjectures.ErdosProblems.«552»

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module

public import FormalConjecturesUtil

Erdős Problem 552

References:

erdosproblems.com/552

[BEFRS89] Burr, S. and Erdős, P. and Faudree, R. J. and Rousseau, C. C. and Schelp, R. H., Some complete bipartite graph-tree Ramsey numbers. Graph theory in memory of G. A. Dirac (Sandbjerg, 1985) (1989), 79-89.

@[expose] public sectionnamespace Erdos552

Determine the Ramsey number $$R(C_4, S_n),$$ where $S_n=K_{1,n}$ is the star on $n+1$ vertices.

A problem of Burr, Erdős, Faudree, Rousseau, and Schelp [BEFRS89].

This problem is #19 in Ramsey Theory in the graphs problem collection.

@[category research open, AMS 5]
theorem erdos_552.parts.i :
    ∀ (n : ℕ),
      SimpleGraph.graphRamsey (SimpleGraph.cycleGraph 4)
        (completeBipartiteGraph (Fin 1) (Fin n)) =
        answer(sorry) := by⊢ ∀ (n : ℕ), (SimpleGraph.cycleGraph 4).graphRamsey (completeBipartiteGraph (Fin 1) (Fin n)) = sorry
  sorryAll goals completed! 🐙

In particular, is it true that, for any $c > 0$, there are infinitely many $n$ such that $$R(C_4, S_n) \leq n + \sqrt{n} - c?$$

@[category research open, AMS 5]
theorem erdos_552.parts.ii : answer(sorry) ↔
    ∀ (c : ℝ), 0 < c →
      Set.Infinite {n : ℕ |
        (SimpleGraph.graphRamsey (SimpleGraph.cycleGraph 4)
          (completeBipartiteGraph (Fin 1) (Fin n)) : ℝ) ≤ (n : ℝ) + Real.sqrt n - c} := by⊢ True ↔
  ∀ (c : ℝ),
    0 < c →
      {n | ↑((SimpleGraph.cycleGraph 4).graphRamsey (completeBipartiteGraph (Fin 1) (Fin n))) ≤ ↑n + √↑n - c}.Infinite
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos552