FormalConjectures.ErdosProblems.«77»

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module

public import FormalConjecturesUtil

Erdős Problem 77

References:

erdosproblems.com/77

[BBCGHMST24] Balister, P. and Bollobás, B. and Campos, M. and Griffiths, S. and Hurley, E. and Morris, R. and Sahasrabudhe, J. and Tiba, M., Upper bounds for multicolour Ramsey numbers. arXiv:2410.17197 (2024).

[CGMS23] Campos, Marcelo and Griffiths, Simon and Morris, Robert and Sahasrabudhe, Julian, An exponential improvement for diagonal Ramsey. arXiv:2303.09521 (2023).

[Er88] Erdős, P, Problems and results in combinatorial analysis and graph theory. Discrete Math. (1988), 81-92.

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

[GNNW24] Gupta, P. and Ndiaye, N. and Norin, S. and Wei, L., Optimizing the CGMS upper bound on Ramsey numbers. arXiv:2407.19026 (2024).

@[expose] public sectionopen scoped Topologynamespace Erdos77

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 find the value of $$\lim_{k\to \infty}R(k)^{1/k}.$$

This problem is #3 in Ramsey Theory in the graphs problem collection.

@[category research open, AMS 5]
theorem erdos_77 :
    Filter.Tendsto (fun k : ℕ ↦ (SimpleGraph.diagonalRamsey k : ℝ) ^ (1 / (k : ℝ)))
      Filter.atTop (𝓝 answer(sorry)) := by⊢ Filter.Tendsto (fun k ↦ ↑(SimpleGraph.diagonalRamsey k) ^ (1 / ↑k)) Filter.atTop (𝓝 sorry)
  sorryAll goals completed! 🐙

-- TODO: Add variants of the problem.

end Erdos77