FormalConjectures.ErdosProblems.«1029»

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module

public import FormalConjecturesUtil

Erdős Problem 1029

References:

erdosproblems.com/1029

[Er93] Erdős, Paul, Some of my favorite solved and unsolved problems in graph theory. Quaestiones Math. (1993), 333-350.

[ErSz35] Erdős, P. and Szekeres, G., A combinatorial problem in geometry. Compos. Math. (1935), 463-470.

[Sp75] Spencer, J., Ramsey's theorem - a new lower bound. J. Combin. Theory Ser. A (1975), 108-115.

@[expose] public sectionopen Filternamespace Erdos1029

If $R(k)$ is the Ramsey number for $K_k$, the minimal $n$ such that every $2$-colouring of the edges of $K_n$ contains a monochromatic copy of $K_k$, then $$\frac{R(k)}{k2^{k/2}}\to \infty.$$

In [Er93] Erdős offers $100 for a proof of this and $1000 for a disproof, but says 'this last offer is to some extent phoney: I am sure that this is true (but I have been wrong before).'

@[category research open, AMS 5]
theorem erdos_1029 :
    Tendsto (fun k : ℕ ↦ (SimpleGraph.diagonalRamsey k : ℝ) /
      ((k : ℝ) * (2 : ℝ) ^ ((k : ℝ) / 2))) atTop atTop := by⊢ Tendsto (fun k ↦ ↑(SimpleGraph.diagonalRamsey k) / (↑k * 2 ^ (↑k / 2))) atTop atTop
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos1029